Reaching Algebra Readiness (RAR). Preparing Middle School Students to Succeed in Algebra - The Gateway to Career Success

REACHING ALGEBRA READINESS (RAR)

Reaching Algebra Readiness (RAR) Preparing Middle School Students to Succeed in Algebra – The Gateway to Career Success

By Tony G. Williams

SENSE PUBLISHERS ROTTERDAM / BOSTON / TAIPEI

A C.I.P. record for this book is available from the Library of Congress.

ISBN 978-94-6091-507-9 (paperback) ISBN 978-94-6091-508-6 (hardback) ISBN 978-94-6091-509-3 (e-book)

Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands www.sensepublishers.com

Printed on acid-free paper

All rights reserved © 2011 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Dedicated to my Uncle Charles, the greatest mathematical and musical mind whom I’ve ever known; and to my siblings, Cheryl and Bruce, who have followed the path of my parents to become outstanding math educators.

TABLE OF CONTENTS

Foreword

ix

Chapter 1. Introduction and Overview

1

Chapter 2. Diagnostic

5

Chapter 3. The Prescriptive Approach

15

Chapter 4. ARA Unit Tests

23

Chapter 5. Interventions

35

Chapter 6. Specific Content Area Interventions – Whole Numbers

47

Chapter 7. Specific Content Area Interventions – Decimals (Unit II)

67

Chapter 8. Specific Content Area Interventions – Fractions (Unit III)

81

Chapter 9. Specific Content Area Interventions – Percents (Unit IV)

99

Chapter 10. Specific Content Area Interventions – Number Concepts and Basic Geometry (Unit V)

113

Chapter 11. Specific Content Area Interventions – Solving Basic Word Problems

131

Chapter 12. Specific Content Area Interventions – Pre-Algebra Concepts

143

Appendix A. Post Algebra Readiness Assessment – (Version 2A) (ARA)

159

Appendix B. Post Algebra Readiness Assessment – (Version 2B–C) (ARA)

163

References

171

vii

FOREWORD

Dear Teacher or Parent, Middle school math occurs at critical phase in the lifespan of student learning. This is the time to close any gaps that exist between the acquisition of a mathematics knowledge base and the skills necessary to succeed in high school and beyond. Essentially, middle school math is both the bridge and the gateway for students. Math skills, that are fortified and expanded in middle school, are needed for completing algebra, which is a deciding factor in determining success for most students. There is a positive correlation between completing algebra and being college bound, highly employable, enrolled in higher math and sciences courses, and confident in higher learning. Students who develop sound fundamental math skills in middle school generally do well in algebra. Problems tend to occur when students have not obtained the necessary skills from their middle school mathematics programs. The National Mathematics Advisory Panel’s (NMAP’s) Final Report (Spring, 2008) presented recommendations addressing critical concerns about mathematics achievement of our nation’s students, particularly in middle schools where the sharp falloff in mathematics achievement in the U.S. begins. A key recommendation of the panel was the need to place increased emphasis on the most critical knowledge and skills leading to algebra. Among ways to accomplish this, the panel suggested streamlining the middle school math curriculum into the most critical topics and the use of drills and exercises to enhance skills that rely on an “automatic (i.e. quick and effortless) recall of facts”. Implementing the panel’s recommendations and developing a practical and useful resource for teachers/parents of middle school students were the motivations for the development of this book. Teachers/parents, in this book, you will find a comprehensive process for ensuring your students/child will receive the tools necessary to succeed in algebra. The Reaching Algebra Readiness (RAR) Process consists of four components: (1) Diagnostic, assessing student’s mastery of the skills needed to take algebra; (2) Prescriptive, developing an individualized plan to address specific math deficiencies; (3) Intervention, utilizing tools and resources (parental involvement, effective teaching strategies, etc), to improve students’ mathematics skills; and (4) Drills and Effective Teachings Strategies, mathematics is a discipline and, simply, there is no way of avoiding practice and drilling in reaching algebra readiness, which can be enhanced significantly by implementing proven effective teaching strategies.

ix

FOREWORD

In this book, the middle school math curriculum has been streamlined to include the most critical topics (whole numbers, decimals, fractions, integers, percents and number concepts, problem solving, and integers and pre-algebra). The diagnostic, prescriptive, and intervention components of the RAR process are all centered on the most critical topic areas. Teaching strategies for the critical math topics are presented in the book and are accompanied with practical drills and exercises that may be used in the classroom. There is no requirement, however, that practice and drills can’t be fun. In this book, every effort has been made to create drills and exercises that are engaging and motivating. The exercises and drills are presented on ready-to-use, black line masters and many include a mini-lesson on the skills involved. Solution keys are also provided for each exercise and drill, along with a grading rubric for all assessments. The exercises and drills can be easily copied and/or made into transparencies for full class instruction and discussion. I hope you find the Reaching Algebra Readiness process and its related materials useful in helping your students acquire the math skills needed for success in algebra and beyond. Sincerely, Tony G. Williams, Ed.D.

x

CHAPTER 1 INTRODUCTION AND OVERVIEW

The Challenge Ahead. The primary challenge facing middle school math teachers is how to ready their students for algebra, which, as supported by overwhelming research, is the gateway to success for many students. There is a substantial correlation between students completing algebra and enrolling in four-year colleges (Horn and Nunez, 2000). Students who complete Algebra1 are twice as likely to graduate from college as students who lack such preparation (Adelman, 1999; Evan, Gray, and Olchefske, 2006). The majority of employees who earn more than $40,000 a year completed algebra in high school (Achieve Inc., 2006). A national poll revealed that two-thirds of the students who completed algebra were well prepared for demands of the workplace (Carnevale and Desrochers, 2003). And yet, there are increasing numbers of students who are not prepared for and fell to successfully complete algebra, as evident by the vast and growing demand for remedial mathematics education among arriving students in four-year colleges and community colleges across the nation. Data show that 71% of America’s degreegranting institutions offer an average of 2.5 remedial courses for skill-deficient students (Business Higher Education Forum, 2005). Overall, these deficiencies are further intensified by factors such as income and race. Research shows that most children from low-income backgrounds enter school with far less knowledge than their peers from middle-income backgrounds, the achievement gap in mathematical knowledge progressively widens throughout their Prek-12 years (NMAP, 2008). However, these achievement gaps can be significantly reduced or even eliminated if low income and minority students increase their success in high mathematics and science courses (Evan, Gray, and Olchefske, 2006). The problem rests not so much in the way algebra is taught as it does in the preparation that students are given. The NMAP final report states that the problems in mathematics learning in the U.S. increase in late middle school before students move into algebra. Results on the National Assessment of Education Progress (NAEP) show near historic highs at Grade 4, yet by Grade 12 no progress on the assessment is evident. In the Trends in Mathematics and Science Study (TIMSS), Grade 8 students in the U.S. did not score as well as students in Grade 4. It is evident that the drop-off is occurring at that crossroads in student achievement – 1 The word “algebra” is capitalized when referring to the particular course sequence of Algebra I and II.

1

CHAPTER 1

middle school. Middle school is the critical stage for closing any gaps between the students’ knowledge base and the math skills needed to succeed in algebra. This is the charge facing today’s middle school mathematics teachers and the parents/guardians of middle school students. It is further compounded by a myriad of factors and issues (economic, political, cultural, and societal) impacting education. School discipline remains a high concern for teachers. There is too much time spent on classroom management and discipline, which takes away from valuable classroom instruction. And now, we’re in an age where teachers must compete for their students’ attention against a number of influences (Internet, television, video games, peer and social pressures.) As daunting as it may seem, the challenge of readying all middle school students for algebra is quite achievable. The book sets out to show you how. Although a great deal of the NMAP’s recommendations must be implemented on a broad and national scale, this book will focus on helping you, as a math teacher/parent, in readying your students/child for success in algebra, which provides unique opportunities and opens academic doors for a vast number of students of diverse backgrounds and unique abilities. There are four primary focuses in the book: diagnostic, prescriptive, implementation, and practice/drill. Each is critically important in addressing any math skills-deficiencies that remain in middle school that will inhibit students for succeeding in algebra. Diagnostic. In order to know what work needs to done, we must first assess, on an individual basis, students’ current level of skills pertaining to algebra readiness. A key question is: What skills are needed for students to be considered ready for algebra? Over the years there has been some debate over the prerequisite skills needed for success in algebra. However, most successful mathematics educators will agree, as supported by a NMAP survey of 743 active algebra teachers, that the essential foundation skills are as follows: (1) accurate and timely computational skills with whole number; (2) proficient computational skills with decimals; (3) proficient computational skills with fractions; (4) general understanding of number concepts including basic percents; (5) understanding of general concepts, proportions, and basic geometry; (6) competency in solving simple word problems; and (7) basic understanding of integers, variables, and simple equations. Chapter 2 in this book addresses the diagnostic component of the RAR process. Included are two Algebra Readiness Assessments, which measure each of the essential foundation skills necessary to reach algebra readiness. (Answer keys and grading scales to measure levels of proficiency are also included.) For practical purposes, the threshold for algebra readiness is set at a 90% rate on the assessments, although there is consideration for students who are within certain ranges of reaching algebra readiness. Depending on their deficiencies and work ethnic/motivation, some students may be given brief remediation, tutoring, and/or refresher activities and retested soon after. Prescriptive. For students who are not quite algebra-ready, we must accurately determine specific math skill-deficiencies in order for us to provide them with the appropriate intervention(s). In Chapter 3, teachers/parents, you will learn how to 2

INTRODUCTION AND OVERVIEW

“drill down” to identify specific deficiencies, then develop a plan accordingly. In much the same way an Individualized Education Program (IEP) is used to plan and document special education services for students with disabilities, teachers will be able to plan and chart areas of interventions needed to help students reach algebra readiness. Using the results of an Algebra Readiness Assessment and the Unit Assessments (Chapter 4) that are presented in the unit, teachers will learn how to develop an Individualized Mathematics Invention Plan (IMIP) to plan a course of intervention for their students and record their progress. Intervention. Chapter 5 explores various interventions and strategies that might be considered in order to help students reach algebra readiness. It focuses on implementing IMIPs utilizing a broad range of interventions including, but not limited to the following: parental involvement, tutoring, drills, worksheets (practice supplements and reinforces), technology (internet and computers), supplemental quizzes, counseling, and study-skills enhancers. The interventions and strategies address specifically the current demands and challenges facing today’s math teachers, particularly in large urban school settings, as they attempt to ready students for algebra. This chapter will show teachers/parents how to address the individual mathematics deficiencies of their students/children, while continuing to implement the requirements of their middle school mathematics curriculum. Using the provided Algebra Readiness Class Tracking Chart, teachers/students will be able to provide targeted supplements, reinforcements, and assessments for students who need them without subjecting other students to a needless and, often uninspiring, review of basic math skills. Drills and Effective Teaching Strategies. In addition to specially designed practice exercises, drills, and tailored quiz/test supplements, Chapters 6–12 will provide valuable tips and strategies addressing the common problems in each of the foundation areas. The teaching strategies presented for each foundation area are based on sound and proven instructional practices. Special attention is given to whole numbers, particularly multiplication, with a focus on both accuracy and timeliness. As much as possible, each drill and exercise was developed to be engaging and motivating while improving student algebra readiness in the critical areas. Each exercise and drill is presented on a ready-to-use, reproducible black line master and many come with a mini-lesson. Also, solution keys are provided, along with a grading rubric for all assessments. Included in Chapters 6–12 are supplemental quizzes and post-assessments for students after they have received adequate instructional interventions.

HOW TO BEST USE THIS BOOK

Far most, this book should be used to assess algebra readiness. Whether you are a teacher, parent, guidance counselor, etc., the algebra readiness assessments will provide valuable information and lead the way in determining next steps. Guidance counselors will find the assessment helpful in making placement decisions, while those teaching an algebra course may use the assessment results to verify proper 3

CHAPTER 1

placement. The assessment can also inform algebra teachers of the extent of any skill deficiencies that a student may have and guide them in plotting a course of action, whether it is a few review exercises or a possible schedule change. Middle school math teachers preparing students for algebra should, within the first few days of school year, administer the Algebra Readiness Assessment and, based on the results, plot an individual course of action for each student. For students who score high (90% or above) on the Algebra Readiness Assessment, consideration should be given to changing their schedules to algebra. For students who are not yet algebra ready, teachers/parents should develop an Individualized Mathematics Intervention Plan (IMIP) and put in place appropriate interventions, strategies, and materials (drills, exercises, supplemental quizzes, tests, etc.) presented in the book. Student progress should be recorded and tracked using the provided Algebra Readiness Class Tracking Chart Provided. The strategies and interventions presented in this book should be reviewed and implemented in a way that works best for you as you work to meet the needs of your students.

4

CHAPTER 2 DIAGNOSTIC

The first step in preparing middle school students to enroll in and successfully complete Algebra is to accurately diagnose their abilities and skills pertaining to those deemed necessary to be considered “ready for algebra.” Diagnosing students’ algebra-readiness for the purpose of providing targeted remediation has been regularly missing from instructional strategies in recent years. Often times, such determinations – if made at all – have been conducted too late to provide students with the individual supports that they need to become algebra ready. No Child Left Behind (NCLB) has led the way in promoting accountability and achievement for students with varying backgrounds, motivational levels, and abilities. However, under NCLB, states vary considerably in assessment standards, requirements, and proficiency determinations. Also, although its intentions are noble, NCLB, as it relates to middle school math, does not focus enough on the skills needed for algebra as suggested by the NMAP. In its final report, the NMAP suggested that state assessments for students through Grade 8 focus on and adequately represent the Panel’s Critical Foundations for Algebra. A key to the success of the Reaching Algebra Readiness system is its proactive approach. The diagnosis of students’ math skills is done to determine which students are algebra-ready and, more importantly, to provide specific individualized assistance to those students who have not yet mastered those essential skills. Remediation of deficient skills will be done without compromising the current middle school mathematics curriculum and state/district assessments requirements. This chapter presents two versions of the Algebra Readiness Assessment (ARA), a 33-problem test designed to measure students’ skills in each of the seven critical areas that are needed to be Algebra ready. The assessments are manual, paper and pencil exams with students showing their work and providing the answers. There are no “choice” answers from which to select. What’s at stake is too important to allow possible chance or guessing to be factored into the outcome determination. (However, given the demands and time restraints facing many teachers, a multiple-choice version of an assessment is also provided.) Included for each assessment is a grading key and scoring rubric. A passing score on the assessment has been set high (90%) because of the essential nature of these mathematics skills in correlation with success in Algebra.

5

CHAPTER 2

The ARA measures students’ mathematics skills on each of the following critical areas: (1) accurate and timely computational skills with whole number; (2) proficient computational skills with decimals; (3) proficient computational skills with fractions; (4) general understanding of number concepts including basic percents; (5) understanding of number concepts and basic geometry; (6) competency in solving simple word problems; and (7) basic understanding of integers, variables, and simple equations. Although there are some ongoing discussions over the perquisite skills needed for success in algebra, nationally, most successful mathematics educators will agree, as supported by a NMAP survey of 743 active algebra teachers, that the skills assessed in the ARA are most critical. The Algebra Readiness Assessment (ARA) should be administered at the beginning of the middle school year in each math course – general math, pre-algebra, or algebra I. Students who pass the ARA (90% or above) should be considered for a schedule change into algebra, if they are not already enrolled in the course. For students scoring in that “gray area” (70 to 89%) on the assessment, it is recommended that a quick review occur followed by retesting. For all others students not passing the assessment, it is important to “drill down” to identify specific skilldeficiencies followed by the development and implementation of an Individualized Mathematics Implementation Program (IMIP) to be used to prepare these students for Algebra. This prescriptive process and the development of the IMIP will be the primary focus in Chapter 3. ALGEBRA READINESS ASSESSMENT (ARA) – (VERSION 1A)

Directions Solve the following problems, showing as much work as needed. Calculators1 are not allowed for this assessment. I. Whole numbers (1) 7,045 + 856 + 91 [1.1] (3) 603 × 97

[1.3]

(2) 9,012 − 823 [1.2]

 (4) 3 831 [1.4]

 (5) 23 825 (remainder)

[1.4]

II. Decimals (6) 2.85 + 3.9 + 11 [2.1] (8) 28.7 × 5.9 [2.3]

(7) 17.3 − 5.04 [2.2]  (9) 2.5 6.25 [2.4]

(10) 43.7 ÷ 100 [2.4]

1 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

6

DIAGNOSTIC

III. Fractions (Simplify answers if needed) (11)

1 4

+

5 6

[3.1]

(12) 4 25 + 1 13

[3.1]

(13)

5 8

−

1 4

[3.2]

7 (14) 3 10 − 2 45

[3.2]

(15)

5 6

·

3 10

[3.3]

(16) 3 15 · 1 14

[3.3]

[3.4]

(18)

÷ 8 13

[3.4]

(20) Write 0.7 as a percent.

[4.2]

(17) 8 ÷

4 5

5 6

IV. Percents (19) Write

1 2

as a percent.

(21) What is 25% of 80?

[4.1] [4.3]

V. Number concepts and basic geometry (22) Find the average of: 23, 47, 110

[5.4] [5.1]

(23) Write 24.3 in words. (24) What is the value of x:

3 2

=

x 6

[5.6]

(25) Compute: 12 − 5 × 2 [5.5] [5.3]

(26) Round 8.015 to the nearest hundredth. (27) What is the perimeter of the rectangular?

[5.7] 3 cm 7 cm

(28) Arrange in order from least to greatest: (a) 0.82 (b) 0.808 (c) 0.18 (d) 0.8

[5.2]

VI. Word problems (29) Students at Central Middle School are required to perform 1800 minutes of community service each year. So far, Keisha has performed 1265 minutes and her best friend, Stephanie, has performed 178 minutes less than her. How many more minutes of community service does Stephanie need perform to meet the school’s requirement? [6.1] (30) Ryan made $44 by raking leaves. If worked a total of 4 hours, how much did he earn for each hour of work? [6.2] 7

CHAPTER 2

VII. Basic integers and pre-algebra (31) What is −10 + 7?

[7.1]

(32) What is −12 ÷ −3?

[7.2]

(33) What is the value of x: 24 = x − 18 [7.3] ALGEBRA READINESS ASSESSMENT (ARA) – (VERSION 1B)

Directions Solve the following problems, showing as much work as needed. Calculators2 are not allowed for this assessment. I. Whole numbers (1) 9,045 + 586 + 23 [1.1] (3) 903 × 87

[1.3]

(2) 8,902 − 697 [1.2]

 (4) 5 915 [1.4]

 (5) 27 943 (remainder)

[1.4]

II. Decimals (6) 3.85 + 13 + 7.9 [2.1] (8) 2.64 × 1.9 [2.3]

(7) 16.3 − 2.04 [2.2]  (9) 2.7 6.21 [2.4]

(10) 83.7 ÷ 100 [2.4]

III. Fractions (Simplify answers if needed) (11)

3 8

+

1 6

[3.1]

(12) 2 34 + 1 12

[3.1]

(13)

8 9

−

2 3

[3.2]

(14) 9 58 − 2 34

[3.2]

(15)

4 7

·

(16) 2 23 · 2 14

[3.3]

÷ 4 12

[3.4]

(20) Write 0.5 as a percent.

[4.2]

[3.3]

5 6

(17) 9 ÷

[3.4]

3 4

(18)

3 4

IV. Percents (19) Write

1 4

as a percent.

(21) What is 20% of 80?

[4.1] [4.3]

2 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

8

DIAGNOSTIC

V. Number concepts and basic geometry (22) Find the average of: 53, 37, 111.

[5.4] [5.1]

(23) Write 3.75 in words. (24) What is the value of n:

2 6

=

1 n

[5.6]

(25) Compute: 10 + 6 ÷ 2 [5.5] (26) Round 8.718 to the nearest tenth.

[5.3]

(27) What is the area of the rectangular?

[5.8] 3 cm 7 cm

(28) Arrange in order from least to greatest: (a) 0.4 (b) 0.05 (c) 0.41 (d) 0.404

[5.2]

VI. Word problems (29) Eli’s goal is to run 13 miles over 3 days. He runs 3 miles on the first day and 5 miles on the second. How many miles must he run on the third day to reach his goal? [6.1] (30) As a babysitter, Lori was paid $5.25 per hour. If she worked 6 hours, how much did she earn? [6.2] VII. Basic integers and pre-algebra (31) What is −10 + (−7)? [7.1] (32) What is 12 · (−3)? [7.2] (33) What is the value of x: 18 + x = 31 [7.3]

ALGEBRA READINESS ASSESSMENT (ARA) – (VERSION 1A–C)

Directions Solve the following problems, showing your all work on separate sheets of paper. Calculators3 are not allowed for this assessment. When you are finished, attach your work to the back of your answer sheet. 3 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

9

CHAPTER 2

I. Whole numbers (1) 7,045 + 856 + 91 [1.1]

(2) 9,012 − 823 [1.2]

(A) 8,992 (B) 8,092 (C) 7,992 (D) 7,982

(A) 8,189 (B) 8,191 (C) 8,199 (D) 7,189

(3) 603 × 97 [1.3]

 (4) 3 831 [1.4]

 (5) 23 825 (remainder)

(A) 508,491 (B) 58,491 (C) 54,491 (D) 58,401

(A) 268 (B) 179 (C) 207 (D) 277

(A) 36 (B) 34 R 43 (C) 35 R 20 (D) 35 R 19

[1.4]

II. Decimals (6) 2.85 + 3.9 + 11 [2.1]

(7) 17.3 − 5.04 [2.2]

(A) 17.75 (B) 6.86 (C) 16.95 (D) 16.75

(A) 12.24 (B) 12.26 (C) 11.09 (D) 12.01

(8) 28.7 × 5.9 [2.3]

 (9) 2.5 6.25 [2.4]

(10) 43.7 ÷ 100 [2.4]

(A) 179.33 (B) 169.53 (C) 1693.3 (D) 169.33

(A) 2.05 (B) 0.25 (C) 2.55 (D) 2.5

(A) 43,700 (B) 4.37 (C) 0.437 (D) 0.0437

III. Fractions (Simplify answers if needed) (11)

1 4

+

5 6

[3.1]

(12) 4 25 + 1 13

7 (A) 1 24

(A) 5 11 15

1 (B) 1 12

(B) 5 38

(C) (D)

10

6 10 11 12

3 (C) 5 15

(D) 5 23

[3.1]

DIAGNOSTIC

(13) (A) (B) (C) (D)

−

5 8 3 4 4 4 4 8 3 8

or 1

5 6

(A)

1 2 1 4 15 30 25 9

(C) (D)

7 (14) 3 10 − 2 45

[3.2]

9 (A) 1 10

(15)

(B)

[3.2]

1 4

(B) (C) (D)

·

[3.3]

3 10

(17) 8 ÷

9 10 3 5 3 1 10

(16) 3 15 · 1 14

[3.3]

1 (A) 3 20

(B) 4 29 (C)4 (D) 3 29 [3.4]

4 5

÷ 8 13

(18)

5 6

(A) 6 25

(A)

5 6

(B) 8 54

(B) 11 35

(C) 10

5 (C) 8 18

(D) 40

(D)

[3.4]

1 10

IV. Percents (19) Write [4.1] (A) 0.5% (B) 5% (C) 20% (D) 50%

1 2

as a percent.

(20) Write 0.7 as a percent. (21) What is 25% of 80? [4.2] [4.3] (A) 7% (B) 70% (C) 0.7% (D) 35%

(A) 25% (B) 32% (C) 20% (D) 15%

V. Number concepts and basic geometry (22) Find the average of: 23, 47, 110

[5.4]

(A) 60 (B) 180 (C) 53 (D) 90 11

CHAPTER 2

[5.1]

(23) Write 24.3 in words. (A) Twenty-four point three (B) Twenty-four and three (C) Twenty-four and three hundredths (D) Twenty-four and three tenths (24) What is the value of x:

3 2

=

x 6

[5.6]

(A) 3 (B) 13 (C) 9 (D) 4 (25) Compute: 12 − 5 · 2 [5.5] (A) 2 (B) 14 (C) −2 (D) 22 [5.3]

(26) Round 8.015 to the nearest hundredth. (A) 8.01 (B) 8.02 (C) 8 (D) 8.015 (27) What is the perimeter of the rectangular?

[5.7] 3 cm 7 cm

(A) 10 cm (B) 21 cm2 (C) 21 cm (D) 20 cm (28) Arrange in order from least to greatest: (a) 0.82 (b) 0.808 (c) 0.18 (d) 0.8 (A) d, c, a, b (B) c, b, d, a (C) c, d, b, a (D) d, a, c, b 12

[5.2]

DIAGNOSTIC

VI. Word problems (29) Students at Central Middle School are required to perform 1800 minutes of community service each year. So far, Keisha has performed 1265 minutes and her best friend, Stephanie, has performed 178 minutes less than her. How many more minutes of community service does Stephanie need perform to meet the school’s requirement? [6.1] (A) 3240 (B) 1535 (C) 535 (D) 713 (30) Ryan made $44 by raking leaves. If he worked a total of 4 hours, how much did he earn for each hour of work? [6.2] (A) $11 (B) $176 (C) $48 (D) $10.25 VII. Basic integers and pre-algebra

(31) What is −10 + 7? [7.1] (A) 3 (B) −3 (C) −17 (D) 17 (32) What is −12 ÷ −3? [7.2] (A) −4 (B) 4 (C) 9 (D) 34 (33) What is the value of x: 24 = x − 18? [7.3] (A) −42 (B) −6 (C) 6 (D) 42

13

CHAPTER 2 ANSWER KEY: ALGEBRA READINESS ASSESSMENT (ARA)

14

CHAPTER 3 THE PRESCRIPTIVE APPROACH

A key component of the Reaching Algebra Readiness program is its prescriptive approach. Middle school is a critical period. It is during the middle-school years that we (math educators, parents who home school) must assess students’ mathematics abilities and skills, and close all gaps that exist between their currents levels and the skills needed to be algebra ready. For students who are ready at the beginning of the school year, we should make every effort to make sure that they are placed appropriately. For students who are not quite algebra-ready, we must accurately determine their skill deficiencies, provide them with necessary remediation and skill enhancements, evaluate and track their progress as recommended by the NMAP’s final report, and provide them with the necessary supports (supplements, parent involvement, tutoring) that will ready them for success. This process must be individualized to the extent possible, given the current challenges and demands placed on middle school math teachers. A major problem in the past has been the “one size fits all approach” for improving students’ computational skills. As a result, students who have acquired certain skills were often disengaged, while students who need the most help received not enough intervention. Although it is clear those students scoring 90% or above on the ARA should be placed in an algebra class if all possible, there is a gray area that must be carefully considered for students who fall just short of 90% and who might receive adequate intervention in time to enroll in an algebra class. This determination should be based on the informed judgment of the math teacher and/or counselor, along with some parental input. Although the suggested gray area is from 70 to 89%, ultimately it’s up to you (the teacher/counselor/parent) to make this determination. The decision should be made on an individual basis, considering student motivation, study habits, discipline/maturity, and parental input. There is no free ride. The criteria that should be strictly adhered to is as follows: The student must be able to receive adequate intervention/remediation, then be retested, scoring 90% or above on an ARA, and then subsequently be enrolled in an algebra course without missing a beat, in other words, without falling behind the other students in the algebra class. If all aspects of the criteria are not met, the decision is simple: The student is not quite algebra ready.

15

CHAPTER 3

This chapter will outline a plan demonstrating how to identify students’ skill deficiencies, plot a course of corrective action, and track progress of students individually. The Algebra Readiness Assessment (ARA) is the initial step in the process. This tool will assist you in determining which students are algebra-ready and which students need interventions (Chapter 5) and, specifically, identifying which contents areas that must be addressed. For students deemed not ready for algebra by ARA, the first step is to “drill down” to determine specific skill deficiencies. The identification of specific mathskill deficiencies is used to develop an intervention plan, called an Individualized Mathematics Invention Plan (IMIP). This plan will be used by teachers/parents as a guide in providing the individualized interventions necessary to get students ready for algebra. In developing an IMIP, the ARA is quite useful but the IMIP should not be developed solely on the results of the ARA, because there are not enough problems for each specific skill to pinpoint deficiencies. In Chapter 4 you will find eight unit tests covering each of the seven critical areas: whole numbers, decimals, fractions, percents, number concepts, word problems, and pre-algebra. (There are two assessments presented for whole numbers, one for whole number operations and one for a rapid recall of multiplication facts.) The unit tests will provide a clearer picture of specific student skill-deficiencies. This is how it works. The ARA is divided into seven units and each problem is coded for a specific skill: I. Whole Numbers 1.1 1.2 1.3 1.4

Adding whole numbers. Subtracting whole numbers. Multiplying whole numbers. Dividing whole numbers.

II. Decimals 2.1 2.2 2.3 2.4

Adding decimals. Subtracting decimals. Multiplying decimals. Dividing decimals.

III. Fractions 3.1 3.2 3.3 3.4

Adding fractions and mixed numbers. Subtracting fractions and mixed numbers. Multiplying fractions and mixed numbers. Dividing fractions and mixed numbers.

IV. Percents 4.1 Convert simple percent to decimal (or reverse). 4.2 Convert simple percent to fraction (or reverse). 4.3 How to find the percent of a number.

16

THE PRESCRIPTIVE APPROACH

V. Number concepts and basic geometry 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Writing decimals in words. Order decimals. Rounding decimals and whole numbers. Find the average (mean). Order of operations (without exponents). Solving simple proportions. Finding the perimeter of a triangle, square, and rectangle. Finding the area of a triangle, square, and rectangle.

VI. Solving Basic Word Problems 6.1 Solving word problems involving addition and/or subtraction. 6.2 Solving word problems involving multiplication and/or division. 6.3 Solving word problems involving more than one computation. VII. Pre-Algebra Concepts 7.1 7.2 7.3 7.4 7.5

Add integers. Subtracting integers. Multiplying integers. Dividing integers. Solving simple equations.

Students with one or more incorrect problems in a particular section of the ARA are to be given the corresponding unit test (see the end of this chapter), starting with the earliest unit in which that student had an incorrect problem(s). (For example, if a student’s first incorrect problem was in Unit III, that student will be given the unit test on fractions.) If a student scores 90% or above on a unit test, the student proceeds to the next unit in the sequence. However, if he or she scores below 90% on the unit test, the student will receive interventions (Chapter 5) on each specific skill of the units in which a problem was missed. For the purpose of developing an IMIP, teachers/counselors/parents may chose to administer to a student the unit test(s) for all the units in which problems were missed; then use the results to plot out of a course of action for each students’ IMIP. After the interventions are completed for a unit, the student is given a post unit test. Interventions for a particular student continue until the student passes the unit test. Upon passing the post unit test (90% or above), the student will focus on each specific skill of the next unit in the sequence in which a problem was missed on the ARA. After all units are successfully completed, students are retested using a Post ARA (see Appendices A and B). Finally, once the student passes the ARA, they are considered algebra-ready and receive periodic reviews to maintain algebra readiness until enrollment in algebra. For students not passing the ARA, the process begins again by reviewing their IMIP and identifying skill-deficiencies with specific units. Here is an example to bring clarity to the process. If Student “A” missed problems on the ARA in sections I, III, V, and VII, the teacher for Student “A” 17

CHAPTER 3

would administer unit tests on whole numbers (operations – 1A and basic multiplication/division facts – 1B ), decimals, number concepts and geometry, and pre-algebra. Let’s say that on the unit tests, Student “A” scored at least 90% for unit I and had adequate efficiency on the multiplication/division facts. However, Student “A” scored below 90% on sections III, V, and VII missing problems from skill indicators 3.2, 3.4, 5.8, and 7.5. As a result, the teacher for Student “A” would develop and implement an IMIP that included interventions on the following: subtracting fractions/mixed numbers (3.2); dividing fractions/mixed numbers (3.4); finding the area of a square and rectangle (5.8); and solving simple equations (7.5). Remember, the student must receive interventions in a skill-deficient area and pass a post unit test on that entire unit before moving to the next identified skill that needs intervention. Once all interventions and post unit tests are successfully completed, Student “A” will be given another opportunity to pass an ARA. And finally, when Student “A” scores 90% on above on a Post ARA, he or she will be deemed ready for algebra and receives periodic practice to keep their skills sharp. However, until a passing score is obtained, the student must continue to be assessed for deficient skills, receive appropriate interventions, and pass appropriate post unit tests. Individualized Math Intervention Plan (IMIP) Each student that lacks algebra-readiness should have an IMIP in place until they reach algebra-readiness. The purpose of the IMIP is to document, manage, and track areas that need intervention for each individual student, as well as to document progress of the students and interventions and efforts used by the teacher. In addition, given the enormous responsibilities and demands placed on math teachers, the IMIP was designed to serve as a tool for math teachers, making the dual tasks of managing curricular requirements and helping students become algebra-ready more manageable. Developing an IMIP is a relatively simple process. The top half of the document contains general information including student information, parent contact information, ARA score(s), and a place for teachers to note contacts (letters, emails, meetings, etc.) with parents. The bottom half of the IMIP is the place where teachers indicate which interventions are needed for the student to become algebraready. Keep in mind that these determinations are made based on the results the Unit Tests administered after the ARA. In developing the IMIP, parental involvement is strongly encouraged and can be an important intervention (Chapter 5), particularly for students with moderate to severe math deficiencies. Parents maybe valuable in determining students’ learning styles and preferences, as well as in determining students’ availability for afterschool tutoring and the use of other interventions, such as computers programs, Internet, private tutoring, etc. Also, it important to note on the IMIP, the type and amount of assistance that parents will be providing to their children at home. Using the same version of the IMIP provided at end of the chapter, an IMIP for Student “A” might look as follows:

18

THE PRESCRIPTIVE APPROACH

19

CHAPTER 3

In looking at the IMIP for Student “A”, it should be obvious that the student needs interventions on subtracting (3.2) and dividing (3.4) fractions and mixed numbers, finding the area of squares, and rectangles (5.8), and solving simple equations (7.5). From the IMIP, you can see that the interventions include supplemental worksheets, extra practice/drills, quiz supplements, and post unit tests. Also, it is noted that the parents of Student “A” will review simplifying fractions with the student four nights each week. The math teacher of Student “A” should update the IMIP as progress is made in each area of intervention. It is important for all teachers to respect the confidentiality of the IMIP document. Each IMIP document should only be shared with the parents, the student, the teacher, and other education professionals. The documents should be stored in a secure place such as a locked file cabinet. At the end of this chapter, there is a blank IMIP form for you to duplicate and get started. Also, to provide clarity of the overall Reaching Algebra Readiness (ARA) process, and for those of you who visional and linear learners, a step-by-step summary and a flowchart model of the process is provided. In Chapter 4, you will find the Unit Tests used in developing the IMIP. Chapter 5 presents an overview of the interventions, and Chapters 6–12 provide teaching strategies and supplements that address each of the critical content areas. QUICK REFERENCE SUMMARY SHEET – ALGEBRA READINESS PROCESS

− Prior to or at the start of the school year, students who are in the appropriate grades will be administered the Algebra Readiness Assessment (ARA). − If feasible, students scoring 90% or above on the ARA will be placed in an algebra course. When placement in algebra is not feasible (e.g. time of year, availability), students remain in their math class and receive periodic reviews to maintain algebra readiness until enrollment in algebra. − For students in the gray area, scoring less than 90% but greater than a score determined at the discretion of the teacher/counselor/parent, these students, after a brief review, are retested using a version of the ARA. Students passing the retest (90% or above) are moved to algebra, if feasible. Students not passing the retest should remain or be placed in the appropriate general mathematics course and begin interventions to become algebra ready. − An Individualized Mathematics Intervention Plan (IMIP) is developed, for students not passing the ARA, by utilizing the results of unit tests. On areas of the ARA that a student missed one or more problems, that student is administered a unit test(s), starting with the earliest unit. If that student scores 90% or more on a unit test, he or she can move on to the next unit in the sequence. (Before developing an IMIP teachers/counselors/parents may choose to administrator to studenst the unit test(s) for all the units in which problems were missed; then use the results to plot out of a course of action for each student on their IMIP.) − If the student scores below 90% on a unit test, they will receive intervention (tutoring, drills/exercises, parental reinforcement, computer interaction, 20

THE PRESCRIPTIVE APPROACH

worksheets, supplemental quizzes, etc.) in the areas where a problem(s) was missed. After a student receives intervention on a unit, a post unit test is given. When the students passes the unit test (90% or above), they move on to the next unit in the sequence. After all units are successfully completed, students are retested using a Post ARA. − Students passing the Post ARA are considered algebra ready and receive periodic reviews to maintain algebra readiness until enrollment in algebra. For students not passing the Post ARA, the process begins again by reviewing their IMIP and identifying skill-deficiencies with specific units. Flow Chart – Reaching Algebra Readiness Process

If students score 90% or more on a unit test, they can move on the next unit in the sequence. If students score below 90%, they will receive intervention in the areas where a problem(s) was missed. After a student receives intervention on a unit, a post unit test is given. After all units are completed, the student in retested using an ARA. 21

CHAPTER 3

22

CHAPTER 4 ARA UNIT TESTS

In this chapter you would find eight unit tests used to make determinations needed to develop Individualized Mathematics Intervention Plans (IMIPs): − − − − − − − −

Unit Test (1A) – Whole Number Unit Test (1B ) – Mastery (speed and accuracy) of Multiplication Tables Unit Test (2) – Decimals Unit Test (3) – Fractions Unit Test (4) – Percents Unit Test (5) – Number Concepts and Basic Geometry Unit Test (6) – Word Numbers Unit Test (7) – Pre-Algebra Concepts

In addition to a unit test on each of the seven critical areas needed for success in algebra, there is a unit test measuring student’s mastery (speed and accuracy) of the multiplication tables. In developing an IMIP, students are given a unit test for each section in which a problem was missed on the Algebra Readiness Assessment (ARA). If the student scores 90% over above the unit test, they move on to the next unit in the sequence. If the student scores below 90%, they will receive intervention on the specific skill indicator in which a problem was missed on the unit. Mastery (Speed and Accuracy) of Multiplication Tables When a student scores below 90% on the Unit IA test on whole numbers, the Unit 1B test on the mastery of multiplication tables should also be administered. This test is used to determine if there is an underlying skill-deficiency that could seriously affect the student’s ability to progress across-the-board in any of critical content areas. When the National Mathematics Advisory Panel’s (NMAP) Final Report (Spring, 2008) recommendations refer to the use of drills and exercises to enhance skills that rely on an “automatic (i.e. quick and effortless) recall of facts, essentially the panel is addressing the importance of mastering (speed and accuracy) multiplication tables which is the most fundamental building block (besides basic addition and subtraction) leading to algebra-ready. Somewhere along the way, perhaps due to confusion over NCTM (National Council of Teachers of Mathematics) standards, we have lost sight over the importance of learning (by whatever means) multiplication tables.

23

CHAPTER 4

The Unit 1B Test, Mastery (speed and accuracy) of Multiplication Tables, consists of 45 simple multiplication and division problems that students must compute, without the use of calculators,1 within 90 seconds. Division problems are included because rapid recall of simple division facts is an outcome of mastering multiplication skills and is essential for success in algebra. The Unit 1B Test was administrated to 63 successful Algebra I students (averaging 70% or above) resulting in mean score of 44.2 with a range from 42–45. The range of the interval was used in determining the following levels of understanding as recommendations, but may be adjusted at the discretion of teacher/counselor. − − − −

Acceptable (42–45) Review and Retest (38–41) Intervention (34–40) Significant Intervention (30 or below)

Following the eight unit tests, at the end of this chapter there are grading keys and a scoring rubrics for each test. In remaining chapters, intervention and specific content area interventions are presented to prepare students for algebra.

RAR – UNIT TEST (1A) WHOLE NUMBERS

Directions Solve to the following problems, showing as much work as needed. Calculators2 are not allowed for this assessment. 1.1 Adding whole numbers (Show all work!) (1) 317 + 53

(2) 10,315 + 5,091

(3) 25,081 + 963 + 46

1.2 Subtracting whole numbers (Show all work!) (4) 9,084 − 285

(5) 1,220 − 388

(6) 10,015 − 1,064

1.3 Multiplying whole numbers (Show all work!) 1 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices. 2 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

24

ARA UNIT TESTS

(7) 253 × 18

(8) 971 × 100

(10) 9,068 × 750

(9) Eight thousand four hundred twenty-five times ninety-seven

(11) 12 × 11

(12) Multiply, then round the product to nearest ten: 25 × 439

[5.1]

[5.3]

1.4 Dividing whole numbers (13) 72 ÷ 8

 (15) 25 7025

(14) 170 ÷ 10

(16) 19,656 ÷ 39

(17) Divide, include reminder in answer.  29 260

(18) Divide, include reminder in answer.  109 3326

RAR – UNIT TEST (1B) MASTERY (SPEED AND ACCURACY) OF MULTIPLICATION TABLES

Directions When you teacher says, “start”, provide as many solutions as you can in the next 90 seconds. 1

7×3=

16

18 ÷ 6 =

31

8×2=

2 3

24 ÷ 8 =

17

8×9=

32

42 ÷ 7 =

4×6=

18

55 ÷ 5 =

33

9×6=

4 5

12 ÷ 2 =

19

4×7=

34

27 ÷ 3 =

3×8=

20

18 ÷ 2 =

35

7×8=

6 7

63 ÷ 7 =

21

5×4=

36

21 ÷ 3 =

5×9=

37

9×3=

64 ÷ 8 =

22 23

36 ÷ 4 =

8

8×8=

38

72 ÷ 8 =

9

11 × 6 =

39

8×4=

28 ÷ 4 =

24 25

50 ÷ 10 =

10

7×7=

40

63 ÷ 7 =

11 12

6×8=

16 ÷ 4 = 9×4=

41 42

12 × 12 =

54 ÷ 9 =

26 27

13

3×5=

28

25 ÷ 5 =

43

5×8=

14 15

48 ÷ 6 =

29

8×6=

30

49 ÷ 7 =

44 45

35 ÷ 5 =

7×9=

81 ÷ 9 =

6×7=

25

CHAPTER 4 RAR – UNIT TEST (2) DECIMALS

Directions Solve to the following problems, showing as much work as needed. Calculators3 are not allowed for this assessment. 2.1 Adding Decimals (1) 26.7 + 0.52

(2) 5.081 + 5.07

(4) 17 + 8.2 + 0.23

(3) 3.8 + 0.25

(5) Add, then round sum to nearest tenth: 7.1 + 0.71 + 0.071

[5.3]

2.2 Subtracting Decimals (6) 10.4 − 7.5

(7) 8.3 − 0.36

(9) 26 − 8.7

(8) 17.93 − 8.8

(10) 7.3 − 0.73

2.3 Multiplying Decimals (11) 6.3 × 5

(12) 2.47 × 0.9

(14) Multiply: seven and eight tenths times seven and eight hundredths

[5.1]

(13) 0.4 × 100

(15) 86.2 × 1.6

2.4 Dividing Decimals  (16) 3 2.01

 (17) 0.5 16.5

(18) 22.94 ÷ 0.31

3 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

26

ARA UNIT TESTS

 (19) 0.43 3.053

(20) Divide, round quotient to the nearest tenth:  0.9 0.225

RAR – UNIT TEST (3) FRACTIONS

Directions Solve to the following problems, showing as much work as needed. Calculators4 are not allowed for this assessment. 3.1 Adding fractions and mixed numbers (1) +

3 4 1 2

(2)

5 9

+8

(3) 4 15 + 6 35

(4) 4 34

(5) Add, then simplify the answer: 8 35

1 6

+ 3 58

+

3.2 Subtracting fractions and mixed numbers (6) −

2 3 1 4

(7)

9 10

−

(8) 10 −

2 3

5 6

5 (10) 12 38 − 4 12

(9) 8 14 − 3 34

3.3 Multiplying fractions and mixed numbers (11)

1 3

×

4 7

(12) 6 ×

2 3

(13) 2 25 × 5

4 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

27

CHAPTER 4

(14)

3 7

× 4 15

(15) 1 18 × 1 79

3.4 Dividing fractions and mixed numbers (16)

3 4

÷

(17) 7 ÷

2 3

(19) 3 14 ÷

1 4

(18) 6 ÷ 1 57

(20) 3 37 ÷ 2 47

2 3

RAR – UNIT TEST (4) PERCENTS

Directions Solve to the following problems, showing as much work as needed. Calculators5 are not allowed for this assessment. 4.1 Convert the percents to decimals (1) 45% = (2) 6% = (3) 5.7% = (4) 100% = (5) 345% = 4.2 Convert the decimals to percents (6) 0.05 = (7) 0.69 = (8) 1.6 = 4.3 Convert the percents to fractions (9) 50% = (10) 3% = 4.4 Convert the fractions to percents (11) (12)

3 4 3 5

= =

5 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

28

ARA UNIT TESTS

4.5 Find the percent of a number (13) What is 25% of 60? (14) Find 3% of 18. (15) 50% of 48 is what?

RAR – UNIT TEST (5) NUMBER CONCEPTS AND BASIC GEOMETRY

Directions Solve to the following problems, showing as much work as needed. Calculators6 are not allowed for this assessment. 5.1 Writing numbers in words (1) Write 12,089,003 in words: (2) Write 8.041 in words: (3) Write two-thousand six hundred twelve and thirty-nine hundredths in standard form: 5.2 List in order from least to greatest: (4) 3.6,

0.63,

0.07,

(5) 0.18,

0.2,

1.3,

(6) 35 ,

2 3,

1 4,

0.604 0.029

1 2

5.3 Rounding (7) Round 8,090,534 to the nearest thousand. (8) Round 6.0832 to the nearest thousandth. 5.4 Find the average of the following group of numbers (Show all work!). (9) 111,

201,

57

6 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

29

CHAPTER 4

(10) 30,

45,

(11) 3.6,

4.4,

15, 12,

105,

75

0.8

5.5 Compute the following: (12) 12 − 6 ÷ 3 + 2 (13) 8 + 20 ÷ (4 + 1) − 3 5.6 Solve for n in the following proportions: (14)

6 n

=

3 4

(15)

7 1

=

n 3

(16)

12 9

=

8 n

5.7 and 5.8 Given the rectangle and square, solve the problems below: (A)

(B) 3 cm

4 cm

5 cm 4 cm 5.9 Answer the questions below: (17) What is the perimeter of Figure A? (18) What is the perimeter of Figure B? 5.10 Answer the questions below: (19) What is the area of Figure A? (20) What is the area of Figure B?

RAR – UNIT TEST (6) WORD PROBLEMS

Directions Solve to the following word problems, showing as much work as needed. Calculators7 are not allowed for this assessment. 7 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

30

ARA UNIT TESTS

(1) Erin ran 2 miles on Monday, 2.5 miles on Wednesday, and 3 miles on Friday. How many miles did she run on all three days? (6.1) (2) For the school’s fundraiser, Barack collected $48.75 and Tomas collected $53.20. How much more did Tomas collect than Barack? (6.1) (3) The Eiffel Tower is about 1063 feet high. The Statue of Liberty is about 305 high, including its foundation and pedestal. If you could put the Statue of Liberty on top of the Eiffel Tower, how high would the two monuments reach? (6.1) (4) The Washington Monument is about 555 feet tall and the Sear Tower in Chicago stands about 1,450 feet. How much taller is the Sear Tower than the Washington Monument? (6.1) (5) This weekend at the mall, Becca bought three sweaters costing $18 each. What was the total cost on all three sweaters? (6.2) (6) Bruce works part-time at the Video Game Center. Last week he worked 12 hours and earned $72. How much did Bruce earn for each hour of he worked? (6.2) (7) Tara baby sits for her neighbor and earns $6.50 per hour. If she works 4 hours, how much will she earn? (6.2) (8) Luther eats about 14,700 calories per week. How many calories does he eat each day? (6.2) (9) Payton wants to run 10 miles over 3 days. He runs 3 miles on the first day and 4 miles on the second. How many miles must he run on the third day to reach his goal? (6.3) (10) Each student at Middletown Jr. High School is required complete 20 hours of community service each year. So far, Melanie has performed 18 hours and her best friend, Hillary, has performed 2 hours less than she. How many more hours of community service does Hillary need to perform to meet the school’s requirement? (6.3) (11) In his Saturday morning teen league, Keyshawn bowled scores of 132, 105, and 153. What was his average score? (6.3) (12) Lance wants to buy a CD that costs $10. If there is a 6% sales tax, what is the total cost of the CD including tax? (6.3)

31

CHAPTER 4 RAR – UNIT TEST (7) PRE-ALGEBRA CONCEPTS

Directions Solve to the following word problems, showing as much work as needed. Calculators8 are not allowed for this assessment. 7.1 Adding Integers (1) −3 + 5 (2) −11 + (−13) (3) 14 + (−23) (4) −12 + (−12) (5) −22 + 17 7.2 Subtracting Integers (6) −23 − 17 (7) −22 − (−19) (8) 15 − (−18) (9) −16 − (−31) (10) 14 − 20 7.3 Multiplying Integers (11) −5 · 8 (12) 3 · (−12) (13) −9 · (−5) (14) 11 · (−4) (15) −7 · (−7) 7.4 Dividing Integers (16) 30 ÷ −6 (17) −24 ÷ −3 (18) −21 ÷ 7 (19) 45 ÷ −9 (20) −18 ÷ −2 7.5 Determine the value of n in each of the following equations: (21) n + 3 = 10

(22) 7n = 49

8 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

32

ARA UNIT TESTS

(23) 9 − n = 15 (25)

n 2

(24) 19 = n − 6

=5

ANSWER KEYS – RAR UNIT TESTS Problem Number 1 2 3 4

5

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1A Answers 370

2 3 4 5 6 7 1E Efficiency Answers Answers Answers Answers Answers Answers Answers 27.22 5/4 or 0.45 * 7.5 2 21 (26) 4 1 1/4 15,406 10.151 8 5/9 0.06 ** 4.45 –24 3 (27) 36 26,090 4.05 10 4/5 0.057 2612.39 1368 –9 24 (28) 5 8,799 25.43 4 11/12 1 0.07, 895 –24 6 (29) 48 0.604, 0.63, 3.6 832 7.881 12 9/10 3.45 0.029, $54 –5 24 (30) 7 0.18, 0.2, 1.3 8,951 2.9 5/12 5% 1/4, 1/2, $6 –40 9 (31) 16 3/5, 2/3 4,554 7.94 7/30 69% 8,091,000 $26 –3 45 (32) 6 97,100 9.13 9 1/6 160% 6.083 2,100 33 8 (33) 54 817,225 17.3 4 1/2 1/2 123 3 15 66 (34) 9 6,801,000 6.57 7 23/24 3/100 54 4 –6 7 (35) 56 132 31.5 4/21 75% 5.2 130 –40 48 (36) 7 10,980 2.223 4 60% 12 $10.60 –36 6 (37) 27 9 40 12 15 9 45 15 (38) 9 17 55.224 9/5 or 0.54 8 –44 8 (39) 32 1 4/5 281 137.92 2 24 21 49 63 (40) 9 504 0.67 9/8 or 6 –5 3 (41) 144 1 1/8 8 R 28 33 28 16 cm 8 72 (42) 9 30 R 56 74 7/2 or 16 cm –3 11 (43) 40 3 1/2 7.1 39/8 or 15 cm2 –5 28 (44) 7 4 7/8 0.3 3/2 or 16 cm2 9 9 (45) 42 1 1/2 7 20 7 9 –6 64 25 5 10 49

∗ Twelve million eight-nine thousand three ∗∗ Eight and forty-one thousandths

33

CHAPTER 4 GRADING SCALE – ARA UNIT TESTS

Number of Problems Missed (Total) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1A – Scores (18) 100 94 89 83 78 72 67 61 56 50 44 39 33 28 22 17 11 6 0

2 (20) 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

3 (20) 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Unit Test 1B – Grading Rubric – – – –

34

Acceptable Review and Retest Intervention Significant Intervention

(42–45) (38–41) (34–40) (30 or below)

4 (15) 100 93 87 80 73 67 60 53 47 40 33 27 20 13 7 0

5 (20) 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

6 (12) 100 92 83 75 67 58 50 42 33 25 17 8 0

7 (25) 100 96 92 88 84 80 76 72 68 64 60 56 52 48 44 40 36 32 28 24 20 16 12 8 4 0

CHAPTER 5 INTERVENTIONS

No Child Left Behind (NCLB), despite criticism over its implementation, unrealistic expectations, and funding, for the most part, has brought about a climate of higher expectations and transparency into mathematics education. Nevertheless, with NCLB, the focus has been primarily on how well schools and local education agencies (LEAs) are doing. Consequently, too much responsibility is placed on teachers, schools, and districts and not enough on students and parents. Further, Annual Yearly Progress (AYP) results, associated with NCLB, are often not timely enough to provide corrective actions to students with skills deficiencies and to those who lack algebra readiness. In the Reaching Algebra Readiness process, teachers, students, and parents share responsibility of students’ preparedness and achievement equally. Teachers assess students’ skills for algebra readiness, develop and implement a course of action in form of an Individualized Mathematics Intervention Plan (IMIP), provide the appropriate interventions for students to achieve readiness. Students provide the effort, time, and commitment necessary to remedy deficient math skill(s). And parents provide the support (practice, drills, reinforcements, private tutoring, etc.) and encouragement necessary for their child to enroll in and succeed in algebra, which has been proven to be a critical determinant of future success. There is enormous pressure on math teachers to prepare their students for statewide and district assessments. Often times, because of time restraints and curriculum demands, teachers are unable to allot a great deal of time on strengthening the basic skills needed to be algebra-ready. However, this can be done successfully. This chapter will provide teachers with the interventions, along with tools and strategies, to do so within their current curriculum. And, of course, supplementing instruction for algebra readiness will only enhance and improve students’ overall scores on all assessments. Although there are many possible interventions that could be considered in preparing students to succeed in algebra, this book will focused only on the research-based interventions that have proven successful: (1) parent involvement; (2) tutoring; (3) drills and practice; (4) counseling and enhancing studiousness; (5) assistive technology; (6) computer guided instruction; and (7) the Internet. Given the current time restraints and demands, teachers must use their best judg-

35

CHAPTER 5

ment in deciding what resources and tools are most effective for their students and add the most value. Parent Involvement Of all possible interventions, parent (guardian) involvement is the most important intervention to all students, particularly for those with moderate to severe skill deficiencies. Research shows that parental involvement has a positive affect on students’ attitudes toward school and/or toward a particular subject area. Parental involvement influences students’ self-concept, classroom behavior, time spent on homework, expectations for the future, absenteeism, motivation, and retention (Cotton and Wikelund, 1989). The need for parent involvement is compounded by the research that shows that middle school is the time when many parents often become less involved in the education of their children. The initial step in involving parents in the RAR process is to notify them of their child’s results on the Algebra Readiness Assessment. It is recommended that notification of parents should be done by a letter sent to the home; and for those students who are not algebra-ready, this should be followed by a phone call inviting them to participate in the development of the Individualized Mathematics Intervention Plan (IMIP) and to strongly encourage them to assist in the process by providing intervention at home. Far most, parental intervention should be in the form of encouragement and support of the efforts of the student and teacher as they work in concert to achieve algebra-readiness. Parents should take the initiative in ensuring that their child participates in any recommended after-school activities such as extra help, tutoring or mentoring. At home, parents should actively be involved in supplemental activities such as drills and practice exercises, and parents must follow-up to make sure assignments and other activities are completed timely by the student. And for parents who can afford it, and for only those parents who can afford it, a recommendation for private tutoring should be a consideration. The following are two sample letters that can be sent home to parents to get the process started: (1) letter notifying parents that their child is deemed algebra-ready; and (2) letter notifying parents of potential math-deficiencies and the need for intervention. (Remember, to obtain permission from school administration before sending home any letter or correspondence.) Sample Parent Letter I Dear

(Parent/Guardian):

Research shows that students who complete algebra are more likely to succeed in college as well as in their chosen careers. Further, research shows that students who have acquired certain critical math skills (fractions, decimals, percents, number concepts, etc.), prior to their enrollment, have a greater chance of succeeding in algebra. I am pleased to inform you that your child, , has passed the Algebra Ready Assessment. Your child has been deemed algebraready and, when appropriate, your child will be enrolled in the course. Until 36

INTERVENTIONS

your child is enrolled, he or she will be given assignments and assessments to maintain their algebra-readiness skills. Being ready for algebra, however, does not guarantee success in the course. Success in algebra will require commitment and hard work from your child, as well as your involvement and support. I’m looking forward to working with you in advancing your child’s mathematics knowledge and skills – a key to their success. If you have any questions, please contact me at the school at (phone number) or by email at . Sincerely,

(Your Name) Sample Parent Letter II Dear

(Parent/Guardian):

I’m looking forward to working with your child and improving their mathematics skills. One of major goals this year is to prepare your child to succeed in algebra; I will need your help to do so. Algebra is very important! Research shows that students who complete algebra are more likely to succeed in college as well as in their chosen careers. Further, research shows that students who have acquired certain critical math skills (fractions, decimals, percents, number concepts, etc.), prior to their enrollment, have a greater chance of succeeding in algebra. Your child has been given an Algebra Ready Assessment and results of the assessment show that your child needs additional work in the following critical areas: . I will be contacting you soon to assist me in developing a plan, an Individualized Mathematics Invention Plan (IMIP), to help your child become ready for algebra. We will look at interventions to assist your child, such as after schools tutoring, drills and practice. I will need you to provide additional practice and reinforcements at home. Parental involvement is critical to your child’s success. By working together, I believe that we can prepare your child for algebra and pave the way for his/her future success. If you have any questions, please contact me at the school at (phone number) or by email at . Sincerely,

(Your Name)

37

CHAPTER 5

Teachers, when talking and meeting with parents, it is vital to emphasize the importance of algebra in their child’s future as well as the importance of their involvement in the process of preparing their child for algebra. In your interactions with parents, keep in mind that the Reaching Algebra Readiness process is not about casting blame on anyone or the current state of (public) education. The RAR process is about identifying student’s math deficiencies (no matter the cause) and providing interventions necessary to provide a remedy. Parents can provide valuable input in developing their child’s Individualized Mathematics Intervention Plan (IMIP). Because the process often occurs early in the school year, teachers have not had a lot of time to get to know each student in terms of their preferences, availability, and learning styles; parents may be able to assist by providing this information. Most parents would like to get involved in their children’s education, but many don’t know how or have an avenue to begin. The development of the IMIP is an ideal way to get them started. On the IMIP, the teacher should note, with agreement of the parent, the parent’s role in the process. Parents should be given a copy of the finished IMIP, so that they can follow along with the process and to serve as a reminder to them of their important responsibility. In most cases, the parents’ role in the process will be to conduct, in the home, drills, extra practice, and the review of assignments. Parents should also be responsible for picking up their children from after school activities such as tutoring and mentoring. When warranted and only will parents can afford to do so, private tutoring could be a consideration. Teachers should provide guidance to parents on how to best drill and provide reinforcements to their child on the specific critical areas that pertain to their individual child. Further, teachers should provide parents with resources such as drills and worksheets (Chapters 6–12 provide specific tools) or direct them to where they can find needed resources. IMIPs should be updated and revised as a student progresses and parents should be kept abreast of the process by notes, emails, calls, etc. Effective and consistent school-home communication is a critical component of the RAR process as well as to student success and achievement. Hopefully, your school district and school already have some strategies in place to promote parental involvement. However, the following are some tips that you may use at the classroom level to foster greater parental involvement: 1. Utilize current technology. If your school or district has a website, use it. If not work with others in your school and community to develop one. (Also, there are some websites available like k12.finalsite.com and www.inetteacher.com.) Use email and information displayed on the website to communicate regularly with the parents as well as students. Parents can easily communicate with teachers via e-mail and/or receive automatic e-mail notification of unexcused absences, missing assignments, failing grades, behavioral concerns, upcoming assignments and exams. It is a wonderful teaching tool, so use it.

38

INTERVENTIONS

2. Make telephone calls to the parents of your students periodically. Have access to their work phone numbers, home phone numbers, cell phone numbers, addresses as well as their email addresses. Make frequent telephone calls to parents to provide updates on student successes as well as the areas where they need improvement (e.g. behavior, homework, test scores, etc.) 3. Send home and have parents sign and return to you as many tests, quizzes, and assignments as possible. Periodically and as needed, send home to parents individualized notes, greetings, and notices (obtain administrative permission when required). 4. Make your classroom inviting and celebrate parent participation. Encourage parents to participate by having special lessons, field trips, presentations, and activities for which they can be invited. 5. Schedule periodic parent-teacher conferences at school. Look upon such conferences as an opportunity for you and the parents to work together as a team for the betterment of the student. Become active in making parents aware of how much their children benefit by their involvement. Show them the research on achievement, self-concept, behavior, expectations for the future, etc. Parent involvement is the most valuable resource in helping students to succeed in their future careers. After School Tutoring As a teacher, any time that you spend providing after school tutoring will pay exponential returns and will actually save you time throughout the school year. It is recommended that each math teacher hold one tutoring session (60–90 minutes) each week. Generally, there are three formats for tutoring: open tutoring, contents specific tutoring sessions, or a combination of both. Open tutoring is when students come to tutoring sessions with varying areas of focus and interest. In content specific tutoring sessions, the teacher generally presents planned strategies and drills to address a specific content area, followed by guided individual practice for students to acquire the desired skill. Because of the individualized nature the RAR process, a combination of the open and content specific tutoring format appears to work best. This approach allows teachers to address a specific content deficiency of a broad number of students as well as assisting students with their current placements as identified by IMIPs. In planning and scheduling afterschool tutoring sessions, the objectives and primary focuses should be made based the teacher’s preferences and their sense of what students’ need. As a guide for RAR, teachers might consider the following time allotment: 1. 2. 3. 4.

Whole number operations and Multiplication Facts – 6 weeks Decimal operations –3 weeks Fractions – 6 weeks Percents – 3 weeks 39

CHAPTER 5

5. Number Concepts and Basic Geometry – 4 weeks 6. Word Problems – 3 weeks 7. Integers and Simple Equations – 5 weeks Teachers are strongly encouraged to recruit volunteers to assist with their after school programs. For talented high school and middle school students with patience and good characters and who have a desire to help others, afterschool tutoring is an excellent vehicle for them to volunteer their services and possible receive community service hours for their efforts. Also, parents with strong math skills, citizens in the community, and adults desiring to be mentors are possible candidates to assist you in tutoring. (Remember, anyone interested in volunteering at your school must be appropriately screened and must successfully complete the approval process required by your school district.) Private Tutoring and Other Parent Funded Assistance Private tutoring and any other interventions that require parents to pay should not be a consideration unless it is absolutely certain that the family can afford such interventions and that the interventions provided, for free, at school and at home will not achieve the desired outcomes. Recommendations for such interventions are usually suggested for students with severe computational deficiencies, perhaps even learning disabilities, which may require intensive professional one-on-one intervention and additional diagnosis. There is a broad range of services available for families that are able and willing to pay for such assistance. These include highly regarded learning centers such as Sylvan and Huntington and a growing number of reparable online services. Regarding finding qualified private tutors, teachers should refer interested parents to a guidance counselor or district official who can provide an approved list of these professionals, which may be include retired teachers, college students, etc. Also, in your district, there may be a list of approved providers to supplement instruction as required for some schools not meeting NCLB’s annual yearly progress (AYP). Because of potential conflicts of interest, it is strongly recommended that math teachers do not provide private tutoring for pay to their own students, particularly during the school year. Drills and Review Exercises Let’s face it: math is a discipline. It requires a certain amount of dedication and commitment in the form of drills and practice in order to develop the knowledge base necessary for further understanding and exploration. Frankly, there is not substitute for drills, practices, and persistent review in developing and maintaining essential math skills. Many traditional mathematics educators argue that this element of disciplined study has been missing in American schools in recent years, resulting in our country falling behind in mathematics and the pure sciences. While many national educators and leaders (legislators, administrators) focus on teaching methods, performance, and skills – which are critically important – as the best way to improve mathematics outcomes, there are those who believe that we should also return to a more fundamental approach to mathematics instruction that includes 40

INTERVENTIONS

drills, exercises, and intense practice to improve results for students in mathematics. Perhaps it’s not fair to compare American public school students to the countries that are excelling in higher mathematics such as Hong Kong and Singapore, which do not have the same freedoms and liberties that we cherish and fight for. On the overhand, we perhaps do not have same levels of discipline and appreciation within our public school systems, which should not be used in any way as an excuse. But can we do a better job? Of course we can. A good place to start is by not only focusing on instructional methods, but also on a renewed commitment to the use of drills and exercises to enhance skills that rely on an “automatic (i.e. quick and effortless) recall of facts’, as recommended by the National Mathematics Advisory Panel’s Final Report (Spring, 2008). There are not many kids that get excited about doing math drills and practice exercises. Most middle school students would probably rather run and hide than to do math drills and practice exercises. (Practice! Practice! Yes, we’re talking about practice!) It doesn’t always have to be like that. Math drills and practice exercises are critical components in helping students acquire the essential skills needed to be ready for algebra. As much as feasibly possible, math activities should be made enticing and fun. Chapters 6 through 12 of this book attempt to do so. These chapters provide a snapshot of the drills and exercises on each of the critical areas that can be used in the classroom, for afterschool tutoring, and at home with parental guidance. The exercises and drills presented are just a start; you will need many more. However, you may rely on the concepts and premises as you research textbooks, the Internet, colleagues’ collections, etc. for others and/or develop your own. The following tips should be considered as you implement drills and exercises into classroom instruction, after school tutoring, and at-home activities: 1. Approach a skill from different perspectives. For example, to reinforce multiplication, division, and other computational skills include speed drills on activities like converting mix numbers to improper fractions (or vice versa), simplifying fractions, and solving simple proportions. 2. Remember that repetition is not bad thing, particularly in learning proper technique and format. For example, doing the same problem over and over again for long division, multiplying multi-digit numbers, and solving basic equations can be helpful in making sequences automatic as well as building student confidence. 3. Use partners and peer. Sometimes students have more explicit ways of conveying ideas and instruction to each other, than we “the trained professionals”. 4. Use visuals and manipulatives like flash cards, playing cards, dice, fractional pies, etc.

41

CHAPTER 5

5. Use technology like Leap Frog devices, interactive educational games, and computer programs that provide drills and focused practice. 6. Don’t be afraid to stress the importance of memorization in the critical foundation areas particularly for multiplication tables and basic computational facts. Also, use memorization for basic equivalences such as 12 = 0.5 = 50%, or 1/3 = 331/3%. Internet, Computers, and Other Technological Devices Educational technology has advanced to the point that there are many excellent interventions – websites, software, and electronic devices – available to assist struggling students. Companies are responding to the growing need of American students in area of mathematics by developing products that help strengthen students’ math skills before they reach higher-level math classes. There are a growing number of computer software programs and Internet websites that can play a significant role in providing interventions that some students need to achieve algebra readiness. For any website and software program used for the purpose of readying students for algebra, the essential criteria is that they address the critical skills by providing drills, exercises, and instruction. In addition, it is important that they are interesting, fun, and user-friendly. Price and licensing are factors that must be considered in selecting any software program. Check with your math supervisor or school administrator to determine which software programs are licensed for use in your school or district. Before recommending any software program to parents, make sure that it is affordable to them without beginning burdensome. Surveying over a 1000 teachers, eSchool News ranked the following as the Best Software for Teaching Pre-algebra: 1. 2. 3. 4. 5. 6.

United Streaming by Discovery Education. Odyssey Math by Compass Learning Inc. Academy of Math by Autoskill International Inc. Bridge to Algebra (Carnegie Learning Inc.). PLATO Interactive Mathematics by PLATO Learning Inc. Destination Math Series by Riverdeep Interactive Learning.

There are hundreds of math education websites worldwide. The key is to determine which address algebra-readiness skills and which are best for the students that you teach. Please keep in mind that, before any website is recommended as an intervention to a parent/child, you should first log on yourself to determine that the website is appropriate for that particular student’s needs. Here are just a few websites that you might consider for technological interventions: 1. 2. 3. 4. 5. 42

http://www.aaamath.com/ http://mathforum.org/dr.math/ http://www.math.com/ http://www.scienceacademy.com/BI/ http://www.mathdrill.com/

INTERVENTIONS

6. 7. 8. 9.

http://www.visualmathlearning.com/ http://www.coolmath4kids.com http://www.free-ed.net/free-ed/Math/PreAlg01/default.asp http://homeschooling.gomilpitas.com/explore/algebra.htm\ #Pre-Algebra 10. http://www.figurethis.org/index.html When considering technology as part of an intervention plan for students to achieve algebra readiness, also be open to electronic devices and educational games, such as Leap Frog’s twist and shout multiplication and division math for students who need to improve their accuracy and speed in the recall of multiplication/division tables. There are several other potentially useful products available including Education Insights’ Math Whiz and Math Shark. LeapFrog’s iQuest system with the grades 6th–8th math cartridge also has good reviews. Quiz/Test Supplements The RAR program is individualized to a great extent through the use of supplemental quizzes/tests. The supplemental quizzes/tests consist of 3–7 problems on a specific critical content area. The supplements were developed to assess students’ progress in a particular critical content area and/or to help students maintain their acquired skills. Different supplements, depending on the student’s status or level, may be attached or added to tests and quizzes on an individualized basis throughout the school year. The use of supplemental quizzes/tests should be completely individualized based on each student’s IMIP and their current unit of review. Each supplement should be graded inclusive of the test or quiz in which it is a part of. The Algebra Readiness Class Tracking Chart presented at the end of this chapter was designed to assist teachers in tracking where students are on the continuum of critical math skills needed to be considered algebra ready. By maintaining and using the chart as a guide, teachers can use the information to determine which supplement is appropriate for each individual student. Using the excerpt below as an example, Mr. Mathe’ would provide the supplements to his students as follows: Student A, Unit III; Student B, Unit I; Student C, Unit V; Student D, comprehensive supplement for maintenance; Student E, Unit II.

The quiz/test supplements may also be administered by themselves. One of the best ways to improve students’ algebra-readiness skills is to give short, frequent quizzes. 43

CHAPTER 5

These supplements should be completed within a reasonable amount of time (4– 10 minutes, pre-determined by the teacher), allowing extra time for students with special needs. When the supplements are administered separately, feedback and a review of the results must be almost immediate to be effective – no later than the next class meeting. Student progress on the quizzes should be charted and the grades/results should be weighted and valued light enough that to not elicit test/quiz anxiety but made significant enough so that students will be encouraged to do their but efforts. You will be amazed by the effectiveness of this technique in terms of building and maintaining student skill levels, and in the building student confidence. Mentoring You may wonder how mentoring may be considered a math intervention. Well, research shows that a mentor can positively impact a student’s self-esteem, selfconfidence, future outlook, study skills, preparation, and focus, all of which can be extremely helpful in improving students’ algebra-readiness skills, particularly for at-risk students. If your school or district has a mentoring program, by all means consider this as a possible intervention for students In addition to providing support and encouragement, mentors can facilitate math drills/exercises and provide tutoring. Be sure that all mentors and volunteers complete the necessary paperwork, receive a background check and screening, and are approved by your school district. Response to Intervention The concept (or latest buzz) of “Response to Intervention” (RTI) has been gaining momentum in education circles in recent years. The idea started as an approach to identify and assist learning-disabled students. However, RTI is now being considered as a practice to keep students from failing in all areas of education, including mathematics. RTI’s basic premise is to detect when a student is on the path to failure then change the path. The first step in RTI is to conduct a formative assessment. Students are evaluated, not to see whether they’re passing or failing, but for the purpose of driving instruction. Some might argue that the Reaching Algebra Readiness (RAR) program is a RTI. However, the RAR program is so specific and detailed in its extent, purpose, interventions, and desired outcomes that RTI might be best considered as one of several possible interventions in the RAR program.

44

INTERVENTIONS

45

CHAPTER 6 SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

In each of remaining chapters, Chapters 6–12, there are exercises and drills for each of the specific content areas. Also, in each chapter there is a brief discussion of effective teaching strategies for that specific content area and, at of end, there are supplemental quizzes and a post unit test. The drills and exercises, as practically possible, are intended to be fun and enticing for students working to achieve algebra readiness. These are just a snapshot of the drills and exercises that will be needed in the classroom, for afterschool tutoring, and for parents to use at home with their children. Many more will be needed. Hopefully, these drills and exercises will serve as a guide as you find others from various resources such as textbooks, workbooks, colleagues, the Internet, research articles, workshops, etc.

WHOLE NUMBERS

If rational number operations are the language of algebra, then whole numbers are its alphabet. Frankly, for middle school students who continue to struggle with whole number operations, there aren’t many options remaining to assist them. The situation is certainly blink, but not totally hopeless. First, it is important to rule out any learning disabilities. (A school counselor or special education coordinator can assist you with any concerns.) For students with an IEP (Individualized Education Program), implementing recommended accommodations such as assistive technology (e.g. calculators, computer) or more time on task, can lead to their success. For all other students, drills and practice exercises are vital along with parent involvement. Adding and Subtracting Whole Numbers Well, for middle school students, what can be said about adding and subtracting whole numbers that hasn’t already? Not much! There isn’t any new research out there or any new revelations. There is no magic wand. Most errors are careless mistakes: not aligning place values; not carrying over values, or not borrowing correctly; and simple computational mistakes. Certainly, students should be taught how to check their answers. For adding, student should know how to check

47

CHAPTER 6

answers by subtracting an addend from the difference and in subtracting by adding the subtrahend and difference to get the minuend.

+

9,730 812 10,542

(10,542 – 812 = 9730)

17,405 – 8,983 8,422 (8,422 + 8,983 = 17,405)

For subtracting, it is suggested for students who are experiencing difficulty, that they complete the borrowing first before they subtract. For example:

Multiplication of Whole Numbers Over the past decade or so, there has been a great deal of misunderstanding over what NCTM (National Council of Teachers of Mathematics) standards say or don’t say (memorization vs. calculator-usage for solving computational problems). As a result of this confusion, there is no surprise that there are large numbers of 6th, 7th, 8th, and even 9th grade students who do not adequately know their multiplication tables. All successful middle school mathematics teachers will tell you that the learning of multiplication tables by their students is critically important for their success in mathematics. Students who do not master this skill (with adequate speed and accuracy without the use of a calculator) – at least for the 1’s through 12’s – are severely limited in their ability to progress in mathematics. Without knowledge of their multiplication facts, students will not be able to master satisfactorily: division; computations involving fractions, percents, and decimals, math applications; algebraic concepts, and the list goes on and on. So as a teacher, what can you do at this late stage? Most successful middle school math teachers will likely tell you that memorization is the most effective method of achieving this. (Let’s face it, memorization is a key fundamental aspect of learning, especially in the early grades, e.g. ABCs). From memorization comes the development of a knowledge base (common understanding), skill development, conceptual understanding, application, and then mastery. To accomplish this, the solution is clear: drill and more drill. Both speed and accuracy must to considered and parental involvement is essential at this late day and time. Also, utilize as many shortcuts and memory aides as possible in teaching your students: − For the 2’s, 5’s, 10’s, and 11’s emphasize their easily memorized multiples (2, 4, 6, 8, . . . , 5, 10, 15, . . . , 10, 20, 30, . . . and 11, 22, 33, 44, . . . )

48

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

− For the 9’s, remember the ten finger method. Hold out your ten fingers, lower the digit corresponding to the number being multiplied by 9, then what’s left is your answer. For example, in 9 × 7 lowering the 7th finger leaves you with 6 fingers to the left and 3 finger to the right. Hence, 63. − Focus separately on the squares (3 × 3 = 9; 4 × 4 = 16; 5 × 5 = 25, . . . ). − Practically, if you were to omit the numbers less than five, the squares, and the 10’s and 11’s, which are relatively simple, there are only six times tables left to memorize and these are usually where the most common mistakes are made: 6 × 7 (knowing it’s the same as 7 × 6) 6 × 8 (or 8 × 6) 7 × 8 (or 8 × 7) 7 × 9 (or 9 × 7) 8 × 9 (or 9 × 8) Here are some other tips/strategies for your consideration: − Insist on students knowing the shortcuts for multiplying by multiples of 10 (10, 100, 1000, etc.) which is to simply move the decimal to right or by placing the same number of zeros in the multiple of 10 at the end of the other factor. (For example, 45 × 100 = 4,500). − Reinforce multiplication skills (speed and accuracy) using other concepts such as converting mix numerals to improper fractions, completing patterns, and solving basic word problems. − Don’t forget to use 2-, 3-, 4-digit multiplication problems to further enhance students’ skills.

Division of Whole Numbers The division of whole numbers is just a basic application of students’ multiplication skills. Reviewing dividing whole numbers should be done in conjunction with the multiplication of whole numbers and, at this point, students should already know the relationship between the two. The teaching strategies are similar and tables, drills, exercises should address both skills as much as possible. As with multiplication, the emphasis on basic division should be on both speed and accuracy. Also, use other concepts to reinforce the division of whole number such as these examples: − Simplifying Fractions and Mixed Numbers: 12 12 ÷ 4 3 = = or 16 16 ÷ 4 4 49

CHAPTER 6

37 2 =5 5 5 − Solving Basic Proportions: 9 n = (9 × 4 = 36 ÷ 6 = 6) 6 4 In addition, insist on students knowing the shortcuts for dividing by multiples of 10 (10, 100, 1000, etc.) which is to simply move the decimal to left according to the number of zeros in the multiple of 10. (For example: 69,100 ÷ 10 = 6,910). Finally, don’t overlook the importance of long division. In addition to improving students’ overall computations skills with all whole number operations, long division can enhance student reasoning as well as promoting discipline and organization. Unfortunately, some teachers shy away from spending sufficient time teaching long divisions, questioning its usefulness in this age of technology. Some argue that long division is too mechanical and dependent on memory. However, there are many math educators who believe that, by lessening the importance of long division, we have hurt the overall computational skill development of many students. Sure, teaching and/or reviewing long division can be a tedious and challenging process. However, it may be math’s “rite of passage” and its benefits could be critical for many students attempting to reach algebra readiness. Here are some tips that might make reviewing long division easier: − Start the review by dividing by single-digit numbers, before moving on to two-digit divisors, then three-digit divisors to emphasize technique and sequence. − Discuss remainders and demonstrate how to check answers by multiplying the quotient times the divisor then adding the remainder to obtain the dividend. − Focus students on getting started; in other words, knowing the first part of the dividend that the divisor will divide into: xx 98 25 2468

25 will not divide into 2 or 24, but it will divide into 246.

 xx 148 509 75,825

509 will not divide in 7 or 75, but it will divide into 758.

− Discuss strategies on for making an “educated guess” to determine correct x 2 xx digits for the quotient. For example, in 41 9072 an “educated guess” for the first digit in the quotient is “2”, because “4” goes in “9” twice. Reassure students that it is okay to use “trail and error” and show them how to make adjustments based on their initial guess. Encourage students to used scrap paper for this process.

50

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

Intervention #1-1

REVIEW 1.1 AND 1.2

Directions Add and subtract the problems below and then use your results to answer the following question: Who was the only President to serve without a Vice President? Place the letters below that holds the place value of your answer. (No calculators please!) I. Add: (1) 801 + 99 S

(2) 9,731 + 2,088 E

(3) 6,256 469 + 38 D

(4) 567 + 67 = R

(5) 38,098 + 2,980 + 897 = J ,

II. Subtract: (6) 382 – 39 H

(7) 17,983 – 8,090 W

(8) 800 – 149 A

(9) 9,421 − 189 = O

5

2

7

6

(10) 5,090 − 2,991 = N,

1

8

4

9

3

2

0

9

2

51

CHAPTER 6

Intervention #1-2

(REVIEW 1.1 AND 1.2)

Directions Add and subtract the problems below. At the bottom, place the number in the box of the corresponding place values. By doing so, you will find the answer the following question: What is the average distance in miles that the earth is from the sun? (No calculators please!) I. Add: (1) 7,049 + 634 =

,



(3) 899,087 + 3,128 + 689 =



(2) 213 + 12 + 2 =



,

II. Subtract: (4) 901 − 72 =



(5) 17,093 − 1095 =

(6) 3,000,000 − 29 =

,

,

III. Compute: (7) 59,086 + (1,897 − 983) =



,

(8) {[100,000,000 + 50,000,000]} − {(20,000,000 + 40,000,000)} =



,

,

 ,   ,    miles

52

,

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

Intervention #1-3

SPEED TRAIL (1.3 AND 1.4) – WHOLE NUMBERS

Directions When your teacher says, “start”, provide as many solutions as you can in the next 90 seconds. 1

20 ÷ 4 =

16

11 × 10 =

31

54 ÷ 6 =

2

3×4=

17

36 ÷ 6 =

32

9×9=

3

18 ÷ 3 =

18

11 × 2 =

33

81 ÷ 9 =

4

6×6=

19

24 ÷ 3 =

34

7×3=

5

14 ÷ 7 =

20

5×7=

35

45 ÷ 5 =

6

8×7=

21

16 ÷ 8 =

36

12 × 12 =

7

24 ÷ 8 =

22

2 × 14 =

37

56 ÷ 7 =

8

5×4=

23

72 ÷ 9 =

38

7×9=

9

25 ÷ 5 =

24

8×6=

39

60 ÷ 6 =

10

6×5=

25

21 ÷ 3 =

40

4×8=

11

22 ÷ 11 =

26

4×4=

41

100 ÷ 10 =

12

3×9=

27

54 ÷ 9 =

42

9×6=

13

63 ÷ 7 =

28

8×5=

43

64 ÷ 8 =

14

13 × 1 =

29

32 ÷ 4 =

44

8×9=

15

28 ÷ 7 =

30

2×9=

45

63 ÷ 9 =

53

CHAPTER 6

Intervention #1-4

ONE-MINUTE DRILL

Directions When the teacher waves the “Go” flag (unveiling the problems on the overhead projector), take you pencils and try to complete the multiplication table within the one minute.

54

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS Invention #1-5

BLANK FORM FOR TEACHERS/PARENTS

Directions Teachers/Parents fill in the top and left side with numbers that your students need the most practice. Allow the student/child one minute to complete, and then review their progress.

55

CHAPTER 6

Intervention #1-6

MULTIPLICATION OF WHOLE NUMBER (1.3)

Directions Find the product of each problem below, and then match your responses with the choices on the right to answer the following question: Which famous mathematician invented Calculus and contributed to the understanding of motion, gravity, and light? (No calculators please!) (1) 29 × 8

(5) 307 × 9

(2) 75 × 6

(3) 41 × 39

(4) 84 × 50

A. 1,599 A. 49,870 C. 89,000 E. 9,876,000

(6) 743 × 20

(7) 701 × 530

(8) 254 × 196

I. 49,784 N. 14,860 N. 2,763 O. 371,530 S. 450 T. 232

(9) 4,987 × 10

8

56

2

(10) 890 × 100

3

9

10

(11) 9,876 × 1000

5

11

4

W. 4200

1

7

6

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

Invention #1-7

DIVIDING WHOLE NUMBERS (1.4)

Directions Divide the following problems and use the quotients to determine the distinguished United States Senator named below. (Show all work and no calculators please!)     (1) 3 111 (2) 5 835 (3) 7 8764 (4) 9 5472 E. 167 K. 110 D. 38 T. 413 E. 607  (5) 25 925

 (6) 37 333

 (8) 319 6699

9

 (7) 14 2912

(9) 39,757 ÷ 89

3

5

10

N. 37 Y. 21 E. 1252 N. 9 D. 208

(10) 110, ÷ 1000

4

6

1

2

7

8

57

CHAPTER 6

Invention #1-8

DIVIDING WHOLE NUMBER (1.4)

Directions Divide the problems below and then use the reminders to complete the BINGO table. There is only one way to win so good luck. (Show all work and no calculators please!)  (1) 4 99

 (2) 9 802

 (3) 15 759

 (4) 19 1870

 (5) 42 889

 (6) 25 7024

(7) 791 ÷ 789

 (8) 89 3504

58

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

Intervention #1-9

WHOLE NUMBERS (1.1, 1.2, 1.3, 1.4)

Find Your Birthday ∗ Take the month you were born (Jan=1, Feb=2, etc.) ∗ Multiply by 4. ∗ Add 13. ∗ Multiply by 25. ∗ Subtract 200. ∗ Add the day of the month you were born. ∗ Multiply by 2. ∗ Subtract 40. ∗ Multiply by 50. ∗ Add the last 2 digits of the year you were born. (i.e. if born in 1971 add 71). ∗ Subtract 10,500. ∗ Now, what do you have?

59

CHAPTER 6

Intervention #1-10

RELATED SPEED DRILLS (1.1, 1.2, 1.3, 1.4)

Directions Complete as many exercises as you can in all tables within the next two minutes. I.

II.

III.

60

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

61

CHAPTER 6 RAR – POST UNIT ASSESSMENT #1 WHOLE NUMBERS

1.1 Adding whole numbers (Show all work!) (2) 24,315 + 6,907

(1) 907 + 69

(3) 75,081 + 863 + 64

1.2 Subtracting whole numbers (Show all work!) (5) 3,550 − 477

(4) 10,784 − 685

(6) 9,051 − 3,461

1.3 Multiplying whole numbers (Show all work!) (7) 452 × 37

(8) 85 × 1000

(10) 7,068 × 590

(11) 13 × 14

(9) nine thousand, three hundred seven times ninety-four

(5.1)

(12) Multiply then round the product to nearest ten: 24 × 739

(5.3)

1.4 Dividing whole numbers (13) 81 ÷ 9 (16) 10,143 ÷ 49

(14) 180 ÷ 10

(17) Divide, include remainder in answer.  23 360

(18) Divide, include remainder in answer.  225 5326

62

 (15) 25 9025

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS RAR – POST UNIT ASSESSMENT (1B) MASTERY (SPEED AND ACCURACY) OF MULTIPLICATION TABLES

Directions When your teacher says, “start”, provide as many solutions as you can in the next 90 seconds. 1

28 ÷ 4 =

16

4×8=

31

36 ÷ 6 =

2

8×3=

17

45 ÷ 5 =

32

5×8=

3

15 ÷ 3 =

18

2 × 13 =

33

24 ÷ 8 =

4

5×6=

19

144 ÷ 12 =

34

4 × 11 =

5

30 ÷ 3 =

20

5×4=

35

44 ÷ 2 =

6

6×3=

21

56 ÷ 8 =

36

8×6=

7

81 ÷ 9 =

22

7×8=

37

54 ÷ 9 =

8

6×6=

23

63 ÷ 7 =

38

7×4=

9

33 ÷ 3 =

24

9×8=

39

24 ÷ 3 =

10

9 × 10 =

25

4÷8=

40

1 × 14 =

11

42 ÷ 7 =

26

9×9=

41

36 ÷ 4 =

12

3×9=

27

72 ÷ 9 =

42

7×9=

13

50 ÷ 10 =

28

8×8=

43

100 ÷ 10 =

14

9×6=

29

49 ÷ 7 =

44

5 × 11 =

15

18 ÷ 6 =

30

7×5=

45

72 ÷ 8 =

Solution Intervention #1-1 A 5

N 2

D 7

R 6

E 1

W 8

J 4

O 9

H 3

N 2

S 0

O 9

N 2

Andrew Johnson 17th President: 1865–1869 (1) 900 (2) 11819 (3) 6763 (4) 634 (5) 41,975 (6) 343 (7) 9893 (8) 651 (9) 9232 (10) 2099 63

CHAPTER 6

Solution Intervention #1-2 92,955,887 miles (1) 7,683 (2) 227 (3) 902904 (4) 829 (5) 15,998 (6) 2,999,971 (7) 58,172 (8) 90,000,000

Solution Intervention #1-3

Solution Intervention #1-4

Solution Intervention #1-5 Answers will vary.

Solution Intervention #1-6 I 8 64

S 2

A 3

A 9

C 10

N 5

E 11

W 4

T 1

O 7

N 6

SPECIFIC CONTENT AREA INTERVENTIONS – WHOLE NUMBERS

Solution Intervention #1-7 T 9

E 3

D 5

K 10

E 4

N 6

N 1

E 2

D 7

Y 8

Senator Edward (Teddy) Kennedy served in the U.S. Senate from 1962 to his death in 2009. Senator Kennedy authored more than 2500 bills throughout his career. Solution Invention #1-8

(1) 24 R3;

(2) 89 R1;

(5) 21 R7;

(6) 280 R24;

(3) 50 R9; (7) 1 R2;

(4) 98, R8; (8) 40 R44.

Solution Intervention #1-9 The answer shows your birthday: / / Month Day Year Solution Intervention #1-10 I.

II.

III.

65

CHAPTER 6

Solution Unit #1 Supplemental Quizzes Quiz IA: (a) 6,870; (b) 731; (c) 23,183; (d) 365 Quiz IB: (a) 9,502; (b) 623; (c) 51,471; (d) 102 Quiz IC: (a) 8,245; (b) 731; (c) 79,392; (d) 201 Quiz ID: (a) 8,613; (b) 687; (c) 20,273; (d) 325

Solution: Post Test Unit IA and IB

GRADING SCALE POST UNIT TEST IA

Grading Rubric – Unit Test IB – – – –

66

Acceptable Review and Retest Intervention Significant Intervention

(42–45) (38–41) (34–40) (30 or below)

CHAPTER 7 SPECIFIC CONTENT AREA INTERVENTIONS – DECIMALS (UNIT II)

Generally, decimals are taught after the introduction of fractions. However, at this level – middle school – we can cautiously assume that students understand that decimals are fractions of a whole. For the purpose of review and remediation, it is more practical for decimal interventions to immediately follow those of whole numbers because essentially they’re the same concept/principle. Adding and Subtracting Decimals With the mastery of adding and subtracting of whole numbers, the addition and subtraction of decimals simply involves aligning decimal points. In doing so, it is important to use examples containing whole numbers and decimals with varying place values. Here are some examples: 4.5 + 10 + 0.48 = 4.5 10.0 + 0.48

9.7 − 0.86 = 9.7 − 0.86

It is recommended that students fill in the blank place values with zeros before adding and subtracting. And for subtracting, it is recommended that students first determine where borrowing is required before subtracting. 8 16 10

4.50 10.00 + 0.48 14.98

9–.70 — − 0.86 8.84

Multiplying and Dividing Decimals For students who have mastered multiplying and dividing whole numbers, multiplying and dividing decimals should be a simple progression. Of course, for multiplying decimals, the key is the proper placement of the decimal in the

67

CHAPTER 7

quotient. Keep it simple: Emphasize counting the number of digits to the right of the decimal in both factors then count that many spaces for right to left in the product for the correct placement of the decimal. Example:

2.59 × 8.3 777 20720 21.497

(2 digits right of decimal) (1 digit right of decimal)

(Total of 3 digits)

As you know, for dividing decimals a key is the proper placement of the decimal in the quotient before division begins. Once again, keep the remediation process simple. Emphasize moving the decimal point in the divisor enough places to the right to make it a whole number, followed by moving the decimal point in the dividend to the right the same number of places, followed by bringing the decimal point straight up for the quotient, and then divide as if dividing whole numbers.  3.5 8.75

=

 2.5 35. 87.5

Here are some additional tips for you to consider when reviewing multiplying and dividing decimals: − Incorporate rounding quotients and products in your review. (This will be addressed further in Chapter 10 on number concepts.) − Show students shortcuts for multiplying and dividing by multiples of 10 (10, 100, 1000) including decimals such as 0.1, 0.01, and 0.001. − Encourage students to check their work by looking back over each step in the computation.

68

SPECIFIC CONTENT AREA INTERVENTIONS

Intervention #2-1

ADDING DECIMALS (2.1)

Directions Add the problems below and then match your responses with the choices on the right to answer the following question: Who was the Greek mathematician who developed a theorem on the right triangle expressing a 2 + b 2 = c2 ? (Show all work and no calculators please!) (1) 2.5 + 0.25 =

(2) 8 + 4.1 =

(3) 17.9 + 2.04 =

A. 12.1 A. 19.94 G. 9.501 H. 11.4

(4) 0.894 + 0.34 =

(5) 0.701 + 8.8 =

(6) 0.4 + 11 =

O. 17.88 P. 5.018 R. 1.234 S. 8.05

(7) 3.8 + 4 + 0.25 =

(8) 11.2 + 1.12 + 0.112 =

(9) 10 + 1.9 + 5.98 =

10

8

T. 2.75 Y. 12.432

(10) 4.01 + 0.008 + 1 =

1

6

2

5

9

4

3

7

69

CHAPTER 7

Invention #2-2

SUBTRACTING DECIMALS (2.2)

Directions Subtract the problems below and then match your responses with the choices on the right to discover the following: The man is recognized as the first African American mathematician; he taught himself Calculus and Trigonometry. (Show all work and no calculators please!) (1) 13.5 − 0.9 =

(2) 0.98 − 0.873 =

(3) 8 − 4.3 =

A. 9.11 B. 0.0111 E. 0.449 E. 0.107 K. 3.7

(4) 6.78 − 4 =

(5) 0.5 − 0.051 =

(6) 10 − 0.89 =

N. 39.7911 N. 12.6 R. 2.78

(7) 40.901 − 1.1099 =

8

70

(8) 0.09 − 0.0789 =

6

1

7

2

3

5

4

SPECIFIC CONTENT AREA INTERVENTIONS

Intervention #2-3

ADDING AND SUBTRACTING DECIMALS (2.1 AND 2.2) (A CALCULATED GREETING)

Directions At first, compute the following problem without a calculator. Once you have finished, checked your answer with a calculator then your calculator upside down for a special greeting:

0.0014 + 7 − 6.79 + 0.562 =

71

CHAPTER 7

Intervention #2-4

MULTIPLYING AND DIVIDING DECIMALS (2.3 AND 2.4)

I. Place the decimal in the correct position for each product. For example:

24.3 × 0.2 4.86

II. Place the decimal in correct position in the quotient that it would be prior to division. •

For example:

72

0.03| 0.21

SPECIFIC CONTENT AREA INTERVENTIONS

Invention #2-5

MULTIPLYING DECIMALS (2.3)

Directions Multiply the problems below then match your products with the choices on the right to determine the branch of mathematics that deals with applications of functions like sine, cosine, and targent. (Show all work and no calculators please!) (1) 3.5 × 7

(2) 0.28 × 0.4

(3) 1.09 × 2.6

(4) 0.07 × 0.08

E. 93.08 G. 0.0007 I. 24.5 M. 0.112

(5) 17.9 × 5.2

(6) 8.09 × 3

(7) 0.076 × 1.2

(8) 0.005 × 0.14

N. 2.834 O. 0.00056 O. 24.27 R. 7.038 R. 285.512

(9) 8.02 × 35.6

(10) 5.01 × 0.402

(11) 0.391 × 18

(12) (0.4)2

T. 0.0912 T. 0.16 Y. 2.01402

12

11

1

8

6

3

4

2

5

7

9

10

73

CHAPTER 7

Intervention #2-6

DIVIDING DECIMALS (2.4)

Direction Divide the problems below then match the quotients to the choices on the right to answer the following question: Who invented the personal transportation called the Segway? (Show all work and no calculators please!)    (1) 0.3 8.52 (2) 0.05 17.5 (3) 4 28.4 A. 2.3 A. 3.6 D. 8.71 E. 20  (4) 0.25 0.0325

 (5) 1.2 4.32

 (7) 0.75 15

 (6) 0.58 1.334

E. 28.4 K. 9.01 E. 0.13 N. 7.1

 (8) 18 162.18

N. 350

 (9) Round answer to the nearest hundredth: 0.7 6.1

9

74

7

5

3

8

6

4

1

2

SPECIFIC CONTENT AREA INTERVENTIONS

Invention #2-7

MULTIPLYING AND DIVIDING DECIMALS BY A MULTIPLE OF 10 (2.3 AND 2.4)

Directions Multiply or divide the following problems by simply moving the decimal point to the left or right. (No calculators please!) Examples:

7.89 × 10 = 78.9

809.1 ÷ 100 = 8.091

I. Multiply. (1) 8.09 × 10 =

(7) 0.009 × 100 =

(2) 17.4 × 100 =

(8) 0.1 × 10 =

(3) 0.895 × 1000 =

(9) 5.2 × 1000 =

(4) 0.6 × 10 =

(10) 0.0908 × 100 =

(5) 0.3056 × 100 =

(11) 365 × 1000 =

(6) 5.9 × 1000 =

(12) 12.789 × 10,000 =

II. Divide. (13) 87.5 ÷ 10 =

(19) 89.98 ÷ 100 =

(14) 305.9 ÷ 100 =

(20) 67,034 ÷ 1000 =

(15) 7,582.08 ÷ 1000 =

(21) 0.09873 ÷ 10 =

(16) 6 ÷ 10 =

(22) 7,908 ÷ 10 =

(17) 2.7 ÷ 100 =

(23) 8 ÷ 100 =

(18) 18.91 ÷ 1000 =

(24) 730.9 ÷ 10,000 =

75

CHAPTER 7

76

SPECIFIC CONTENT AREA INTERVENTIONS RAR – POST UNIT ASSESSMENT #2 DECIMALS

Directions Solve the following problems, showing as much work as needed. Calculators are not allowed for this assessment. 2.1 Adding Decimals (1) 45.7 + 0.92

(3) 2.8 + 0.52

(2) 3.075 + 1.24

(4) 18 + 9.2 + 0.45

(5) Add, then round sum to nearest tenth: 8.2 + 0.82 + 0.082

(5.3)

2.5 Subtracting Decimals (6) 11.3 − 5.7

(9) 32 − 9.7

(7) 9.3 − 0.25

(8) 18.03 − 7.7

(10) 3.7 − 0.37

2.6 Multiplying Decimals (11) 5.9 × 7

(12) 2.07 × 0.6

(14) Multiply: sixth and four tenths times three and two hundredths

(13) 0.8 × 100

(15) 76.2 × 3.4 (5.1)

77

CHAPTER 7

2.7 Dividing Decimals  (16) 3 5.10

 (17) 0.5 17.5

 (19) 0.59 1.357

(18) 9.338 ÷ 0.23

(20) Divide, round quotient to the nearest  hundredth: 0.6 0.135

Solution Intervention #2-1 P 10

Y 8

T 1

H 6

A 2

Solution Intervention #2-2 Banneker, Benjamin (1) 12.6 (2) 0.107 (3) 3.7 (4) 2.78 (5) 0.449 (6) 9.11 (7) 39.7911 (8) 0.0111 Solution Intervention #2-3 O. 7734 Greeting: hELLO Solution Intervention #2-4 I. (1) 0.036 (2) 86.5 (3) 0.0004 (4) 86.4 (5) 0.2958 (6) 7.0126 (7) 637.56 (8) 0.0000075 (9) 1.64592 (10) 0.04

78

G 5

O 9

R 4

A 3

S 7

SPECIFIC CONTENT AREA INTERVENTIONS

II.

Solution Intervention #2-5 T 12

R 11

I 1

G 8

O 6

N 3

O 4

M 2

E 5

T 7

R 9

Y 10

Trigonometry is a branch of mathematics that deals with applications of functions like sine, cosine, and tangent. Solution Intervention #2-6 D 9

E 7

A 5

N 3

K 8

A 6

M 4

E 1

N 2

In 2001, Dean Kamen created a personal transportation vehicle called the Segway. Solution Intervention #2-7 (1) 80.9 (2) 1740 (3) 895 (4) 6 (5) 30.56 (6) 5900 (7) 0.09 (8) 1 (9) 52,000 (10) 9.08 (11) 365,000 (12) 127,890 (13) 8.75 (14) 3.059 (15) 7.58208 (16) 0.6 (17) 0.027 (18) 0.01891 79

CHAPTER 7

(19) 0.8998 (20) 67.034 (21) 0.009873 (22) 790.8 (23) 0.08 (24) 0.07309 Solution Unit #2 Supplemental Quizzes Quiz 2A: (a) 27.32; (b) 17.37; (c) 114.72; (d) 3.65 Quiz 2B: (a) 94.94; (b) 69.32; (c) 51.471; (d) 10.2 Quiz 2C: (a) 26.57; (b) 7.31; (c) 79.392; (d) 20.1 Quiz 2D: (a) 16.69; (b) 75.72; (c) 202.73; (d) 32.5 Solution Unit 2 Post Assessment (1) 46.62 (2) 4.315 (3) 3.32 (4) 27.65 (5) 9.1 (6) 5.6 (7) 9.05 (8) 10.33 (9) 22.3 (10) 3.33 (11) 41.3 (12) 1.242 (13) 80 (14) 19.328 (15) 259.08 (16) 1.7 (17) 35 (18) 40.6 (19) 2.3 (20) 0.23

GRADING SCALE UNIT 2 POST ASSESSMENT

80

CHAPTER 8 SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Hopefully, it is safe to assume that students at this level (middle school) have an understanding of what a fraction is and its relationship to a whole. Given that, initially it is still essential that all students have the following skills before moving on to operations with fractions: (1) students must be able to simplify common fractions by dividing both the numerator and denominator by a common factor; (2) students must be able to rewrite a fraction using a different denominator; (3) students must be able to convert an improper fraction to a mixed number; and (4) students must be able to convert a mixed number to an improper fraction. Spend the time necessary for students to master those skills before reviewing addition, subtraction, multiplication and division of fractions and mixed numbers. Adding Fractions Of course, in order to add and subtract fractions, you must have a common denominator. The key problems that students face are rewriting fractions using a common denominator and properly simplifying fractions. A helpful tip might be to focus not so much on the least common denominators (or least common multiples) as common denominators (or common multiples). In the end, by continuing to simplify, the answers will be the same. For example, in adding, 34 + 16 , you will get the same answer rather use 24 or 12 (LCD) as your common denominator. 3 18 = 4 24 +

1 4 = 6 24 22 11 = 24 22

3 9 = 4 12 +

1 2 = 6 12 11 22

For simplifying fractions, remember to remind students that, by continuing to divide both the numerator and denominator by a common factor, in the end you would

81

CHAPTER 8

get the same result as dividing initially by the greatest common factor (GCF). For example, 18 18 ÷ 2 = 9 ÷ 3 3 = = 48 48 ÷ 2 = 24 ÷ 3 8

18 18 ÷ 6 3 = = 48 48 ÷ 6 8

Subtracting Fractions The principles for subtracting fractions and mixed numbers are very similar to those of addition. Obviously, the borrowing aspect presents the greatest challenge. However, as a teacher, you can make this concept easier to understand by demonstrating “one” as a fraction. As a fraction, “one” can be written in an infinite number of ways. Any number over itself (2/2, 7/7, 50/50, etc.) is equal to one. Depending on the common denominator that you desire, “one” can be made to order. For example, in 9 − 3/7 it would be most convenient to use “one” in the form of 7/7. Thus, “9” may be written as 8 7/7. With common denominators, we with can now subtract as follows: 8

7 7 3 7

− 8

4 7

Let us use another example, 6 3/5 − 2 4/5, to illustrate this concept. We cannot subtract 4/5 from 3/5 so we borrow “one” from the whole number “six” giving it to the 3/5, which now gives us 1 3/5. Since our common denominator is “five”, the “one” we borrowed is in the form of 5/5, thus 5/5 plus 3/5 gives us 8/5. Now we are able to subtract:

82

6

3 5

=

5

8 5

− 2

4 5

=

2

4 5

3

4 5

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Here one final example:   2 8 12 8 = 7 =6 + = 3 12 12 12 − 5

3 15 = 5 = 4 20

6

20 12

5

15 20

1

5 1 =1 20 4

Multiplying and Dividing Fractions Generally, multiplying and dividing fractions and mixed numbers is less problematic than addition and subtraction. However, for students that do not fully understand the concepts, multiplication and division can cause frustration. Most major problems occur when students confuse the concepts of adding and subtracting fractions with those of multiplying and dividing fractions which do not require the use of a common denominator. Other problems occur with students have not mastered their multiplication tables and other skills such as converting mixed numbers to improper fractions and simplifying fractions. Of course for multiplying and dividing mixed numbers, students must first be able to convert these to improper fractions. In your review of multiplication, cover both options: traditional (numerator times numerator and denominator times denominator); and cross canceling when able, such as the following example:

As you know, when it comes to dividing fractions, we are simply multiplying by the reciprocal of the divisor. The keys, from an instructional prospective, are to help students identify the divisor (the fraction that we’re dividing by) and to help fully understand to concept of this reciprocal. This should be accomplished using mixed numbers and whole numbers in addition to common fractions. Here are some examples of reciprocals: 2 3 3

=

3 2

1 16 = 5 5 5 =

1 5 83

CHAPTER 8

Finally, let us put it all together using this example:

84

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Intervention #3-1

FRACTION FOUNDATIONS (3.0)

Directions Complete each section that contains the foundations for performing operations with fractions. I. Rewrite each fraction with the denominator provided.

II. Rewrite each improper fraction as a mixed number.

III. Simplify each fraction.

85

CHAPTER 8

IV. Write each mixed number as an improper fraction.

V. Write the reciprocal of each number.

86

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Intervention #3-2

ADDING FRACTIONS (3.1)

Directions Add the problems below then match the sums to the choices on the right to answer the following question: Who are the two co-founders of the search Google? (Simplify all answers when necessary and show all necessary work!)

87

CHAPTER 8

Intervention #3-3

SUBTRACTING FRACTIONS (3.1)

Directions Subtract the common fractions from the whole numbers then draw a line to the correct answer on the right. (Show all work). Example:

5 −

1 = 4

4

4 4

−

1 4

4

3 4 2 6 1 3

(1) 6 2 − 5

(A) 8

(2) 7 1 − 6

(C) 10 (D) 5

2 5

(3) 8 1 − 2

(E) 7

1 2 3 5

(4) 10

(G) 6

− (5) 8 4 − 6

88

(B) 7

(F) 5

3 7

(H) 9 (I) 8

1 2

5 6 4 7

4 7

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Intervention #3-4

SUBTRACTING FRACTIONS (3.1)

Directions Subtract the common fractions and mixed numbers below then match your answers to the choices on the right to answer the following question: Which President appeared on the $500 bill?

89

CHAPTER 8

Intervention #3-5

MULTIPLYING AND DIVIDING COMMON FRACTION (3.3 AND 3.4)

Directions Multiply or divide the common fractions below. Once you a finish, match the answers that are the same. There should be five pairs of identical answers. I. Multiply, cross cancel whenever possible. (1)

3 2 · 4 9

(6)

11 15 · 3 22

(2)

4 5 · 5 8

(7)

(3)

5 5 · 8 9

25 3 · 24 10

(4)

5 ·2 8

(5)

7 7 · 18 28

1 6

(9)

5 3 · 8 10

(10)

(8) 6 ·

1 7 · 4 18

II. Divide by multiplying by the reciprocal of the divisor. Cross cancel whenever possible. (11)

5 10 ÷ 9 27

(12)

(15)

8 5 ÷ 15 4

(16) 8 ÷

(18)

7 35 ÷ 36 24

90

9 3 ÷ 32 2

(19)

4 5

(13)

(17)

17 17 ÷ 31 31

1 ÷6 6

(14)

7 7 ÷ 18 9

(20)

25 5 ÷ 81 27

20 4 ÷ 21 7

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

Intervention #3-6

MULTIPLYING COMMON FRACTIONS AND MIXED NUMBERS (3.4)

Directions Multiply the problems below and then use the products to complete the BINGO table. There is only one way to win so good luck. (Show all work and simplify answers when possible.) I. Multiply. (1) 1

2 5 · 3 10

(5) 2

2 2 · 3 5

(9)

26 7 ·1 35 13

(2) 2

(6) 2

2 7 · 3 8

2 3 · 11 8

(10)

(3) 6 ·

(7) 5

1 3 ·5 19 7

5 12

(4) 2

1 1 ·5 4 7

(11) 9

1 13 ·1 3 14

(8) 9

1 25 ·1 3 26

1 3 ·1 2 4

(12) 5

5 1 ·2 7 10

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CHAPTER 8

Intervention #3-7

DIVIDING COMMON FRACTIONS AND MIXED NUMBERS (3.4)

Directions Divide the common fractions and mixed numbers below then match your quotients to the choices on the right to answer the following question: Who invented Logarithms problems? (Simplify your answers whenever possible!) 3 = 5

(2) 3

3 ÷9= 8

(3)

2 1 ÷2 = 3 10

(5) 4

4 18 ÷ = 5 25

(6) 5

(1) 18 ÷

(4) 6

9 2 ÷ = 11 22

5 12 1 E. 4 2 A.

1 3 ÷3 = 7 7

H. 1 I.

(7) 2

2 1 ÷5 = 9 3

(10) 5

(8)

18 5 ÷1 = 35 49

(9) 2

5 2 ÷4 = 6 3

1 2

3 8

17 28 2 N. 6 3 J.

3 ÷7= 5

4 5 7 O. 15 11 P. 3 63 N.

R. 30

9

92

8

6

5

10

7

4

2

3

1

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

93

CHAPTER 8 RAR – POST UNIT ASSESSMENT (3) FRACTIONS

Directions Solve to the following problems, showing as much work as needed. Calculators are not allowed for this assessment. 3.1 Adding fractions and mixed numbers. (Simplify answers where possible.) 5 8 1 + 4

(1)

(4) 5 +

(2)

3 4 5 6

7 +9 11

1 7 3 +8 7

(3) 3

3 5 5 + 4 8

(5) 10

3.2 Subtracting fractions and mixed numbers. (Simplify answers where possible.) 3 4 1 − 3

(6)

1 8 3 − 2 8

(9) 8

(7)

7 2 − 10 3

(8) 9 −

(10) 11

7 8

3 7 −4 8 12

3.3. Multiplying fractions and mixed numbers. (Simplify answers where possible.) 1 5 × 2 7 5 1 (14) × 5 7 4 (11)

94

(12) 8 ×

3 4

5 2 (15) 2 × 2 8 7

5 (13) 4 × 6 6

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

3.4. Dividing fractions and mixed numbers. (Simplify answers where possible.) 5 3 ÷ 6 5 3 2 (19) 2 ÷ 8 3

1 5 7 4 (20) 7 ÷ 3 9 9

(17) 8 ÷

(16)

(18) 22 ÷ 1

5 6

Solution Intervention #3-1 (3.0) (1) 3;

(2) 12;

(3) 30;

(4) 64;

(5) 21;

(6) 2;

(7) 25;

(8) 15;

2 1 1 (13) 4 ; (14) 4 ; (15) 5; (16) 5 ; 3 2 4 1 5 3 8 3 (17) 1 ; (18) 6; (19) 2 ; (20) 2 ; (21) 2 ; (22) 3; (23) 1 ; 17 7 10 11 14 3 3 1 1 4 2 2 3 (24) 7 ; (25) ; (26) ; (27) ; (28) ; (29) ; (30) ; (31) ; 5 4 2 8 9 5 5 5 1 1 5 5 4 16 19 (32) ; (33) ; (34) ; (35) ; (36) ; (37) ; (38) ; 2 3 12 8 5 3 4 23 39 37 72 53 25 37 (39) ; (40) ; (41) ; (42) ; (43) ; (44) ; (45) ; 12 8 4 7 6 9 6 37 71 145 4 8 1 (46) ; (47) ; (48) ; (49) ; (50) ; (51) 7; (52) ; 5 6 12 3 9 4 6 11 20 3 1 (53) ; (54) ; (55) ; (56) ; (57) ; (58) 10; 5 12 17 28 7 8 7 (59) ; (60) 43 18 (9) 18;

(10) 21;

(11) 3;

(12) 42;

Solution Intervention #3-2 (3.1) L 19

A 5

R 16

E 2

R 4

R 8

Y 1

P 17

A 6

G 12

E 3

B 14

R 7

I 15

and S 18

G 9

E 13

Y 11

N 10

In 1996 Larry Page and Sergey Brin co-founded the search engine Google.

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CHAPTER 8

Solution Intervention #3-3 (3.2) 3 5 5 (2) G, 6 6 1 (3) E, 7 2 4 (4) H, 9 7 1 (5) B, 7 3 (1) F, 5

Solution Intervention #3-4 (3.2) President William McKinley is printed on the US $500 bill! It was last printed in 1934. Solution Intervention #3-5 1 1 25 (1) ; (2) ; (3) 6 2 72 1 7 1 (4) 1 ; (5) ; (6) 2 4 72 2 5 3 (7) ; (8) 1; (9) 16 16 7 1 3 (10) ; (11) 1 ; (12) 72 2 16 1 2 32 (13) ; (14) 1 ; (15) 36 3 75 1 2 (16) 10; (17) ; (18) 2 15 2 (19) 1; (20) 1 3 Matching pairs: (2, 17); (5, 10); (9, 12); (8, 19); (14, 20). Solution Intervention #3-6 1 1 1 (1) ; (2) 2 ; (3) 2 ; 2 3 2 1 1 9 (4) 4 ; (5) 1 ; (6) ; 2 15 11 5 1 (7) 27; (8) 16 (9) 1 ; 8 7 96

SPECIFIC CONTENT AREA INTERVENTIONS – FRACTIONS (UNIT III)

2 (10) ; 7

(11) 18

4 ; 13

(12) 12

Solution Intervention #3-7 J 9

O 8

H 6

N 5

N 10

A 7

P 4

I 2

E 3

R 1

Logarithms were invented by John Napier, about 400 years ago, to make the multiplication and division of large numbers easier. The logarithm of a number is simply the exponent that indicates the power to which a base must be raised to produce that number. For example, log2 8 = 3 (read the logarithm of 8 to the base 2 equals 3) because two raised the 3rd power equals 8. 0.1. Solution Unit #3 Supplemental Quizzes 5 1 1 Quiz 3A: (a) 4 ; (b) 5 ; (c) 8; (d) 6 5 4 13 1 1 Quiz 3B: (a) 5 ; (b) 3 ; (c) 23; (d) 18 6 12 17 3 Quiz 3C: (a) 1 ; (b) 8 ; (c) 9; (d) 5 24 8 13 5 4 Quiz 3D: (a) 7 ; (b) 4 ; (c) 4; (d) 1 18 8 121 Solution RAR – Post Unit Assessment (3) 7 7 (1) ; (2) 9 ; 8 11 1 3 (8) 8 ; (9) 5 ; 8 4

4 ; 11 19 (10) 6 ; 24 (3) 11

7 ; 12 5 (11) ; 14 (4) 6

(5) 15

9 ; 40

(12) 6;

(6)

5 ; 12

(13) 29;

1 ; 30 3 (14) 3 ; 4 (7)

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(15) 6;

(16) 1

7 ; 18

(17) 40;

(18) 12;

(19) 3

9 ; 16

(20) 2

GRADING SCALE UNIT 3 POST ASSESSMENT

98

8 31

CHAPTER 9 SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

In preparation for algebra, obviously students need to know what a percent really is, which put simply, is a ratio of parts out of a 100. In addition, students need to be able to convert a percent to a decimal and fraction and, at a minimum; students should be able to find the percent of a number. When it comes to converting percents to decimals/fraction, quite frankly there are some common ones that all students should know by memory. These are the following:

Decimal

Percent

Fraction

0.5 0.25 0.75 1 0.2 0.60 0.33. . . 0.66. . .

50% 25% 75% 100% 20% 60% 33 1/3% 66 2/3%

1 2 1 4 3 4

1 1/5 3/5 1/3 2/3

Students should know methods for converting percents to decimal and fractions. As you know, because a “percent” means out of 100, converting a percent to a decimal or fraction is done simply by dividing by 100. In your review, remind students that converting a percent to a decimal can be done by moving the decimal point two places to the left, which is the same as dividing by hundred. In converting percents to fractions, remind students that they can multiply by one over hundred (1/100) which is also the same as dividing by hundred. Here some examples of both: 30% = 0.30 or

30 1 = 100 3

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CHAPTER 9

250% = 2.5 or

250 1 =2 100 2

1 46 1 46 23 5 %= · = = 9 9 100 900 450 Be sure to remind students that converting decimal/fractions back to percents can be done by doing the opposite, multiplying by 100. For example, 0.4 = 40% and 7/8 = 7/8 × 100 = 87.5%. And finally, students should be able to find the percent of a number by understanding that this is accomplished by converting the percent to a decimal or fraction then multiplying. Here are a couple of examples: (a) 25% of 16 = 0.25 × 16 = 4 (b) 33 1/3% of 60 = 1/3 × 60 = 20

100

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

Intervention #4-1

CONVERTING PERCENTS TO DECIMALS (4-1)

The word percent means parts of a hundred. In converting a percent to a decimal, you are actually dividing by 100. Perhaps, in many cases, the easiest method of doing this is to simply move the decimal point two places to the left (which is the same as dividing by 100). Directions Convert the following percents to decimals: (A) 9%

(B) 67%

(C) 0.2%

(D) 600%

(E) 12 %

(F) 9.45%

(G) 0.006%

(H) 100%

(I) 50%

(J) 3 1/3%

(K) 6.7%

(L) 1%

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CHAPTER 9

Intervention #4-2

CONVERTING DECIMALS TO PERCENTS (4-2)

Remember, the word percent means parts of hundred. Therefore to convert a decimal to a percent, just multiply by hundred. Perhaps, in many cases, the easiest method of doing this is to simply move the decimal point two places to the right (which is the same as multiplying by 100). Directions Convert each decimal to a percent. (1) 0.12 (2) 0.4 (3) 0.045 (4) 2.9 (5) 6 (6) 0.0103 (7) 30 (8) 14.9 (9) 0.2 (10) 0.75 (11) 1 (12) 0.09 (13) 100 (14) 1.009 (15) 0.0002

102

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

Intervention #4-3

CONVERTING PERCENTS TO FRACTIONS (4-3)

Remember, the word percent means parts of a hundred. In converting a percent to a fraction, you are actually dividing by 100. Perhaps, in many cases, the easiest method of doing this is to simply divide by 100 or place 100 in your denominator. Don’t forget to simplify your answers where possible. Directions Convert the percents below to fractions then match your answers with the choices on the right to answer the following question: What unit is used to measure the speed of microprocessors, representing one million cycles per second?

103

CHAPTER 9

Intervention #4-4

CONVERTING FRACTIONS TO PERCENTS (4-4)

Remember, the word percent means parts of hundred. Therefore to convert a fraction to a percent just multiply by hundred. Perhaps, in many cases, the easiest method of doing this is to simply multiply by 100 1 then simplify. Directions Convert the fractions to percents below then match your answers with the choices on the right to answer the following question: Who was the business man with an idea that originated the iPod?

104

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

Intervention #4-5

PERCENTS (4.1, 4.2, 4.3, 4.4)

Directions In each row, there is one incorrect number. Identify the incorrect number in each row.

105

CHAPTER 9

Intervention #4-6

FINDING THE PERCENT OF A NUMBER (4-5)

To find the percent the percent of a number you must first convert the percent to either a decimal or fraction then multiply. Directions Find the percent of the numbers in the problems below then match your answers with the choices on the right to answer the following question: Who was the French mathematician who invented the first calculating device?

106

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

107

CHAPTER 9 RAR – POST UNIT TEST (4) PERCENTS

Directions Solve to the following problems, showing as much work as needed. Calculators are not allowed for this assessment. (4.1) Convert the percents to decimals (1) 35% = (2) 9% = (3) 5.7% = (4) 0.08% = (5) 209% =

(4.2) Convert the decimals to percents (6) 0.4 = (7) 0.79 = (8) 3.2 =

(4.3) Convert the percents to fractions (simplify when possible) (9) 60% = (10) 5% =

(4.4) Convert the fractions to percents (11)

108

4 = 5

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

(12)

17 = 20

(4.5) Find the percent of a number (13) What is 40% of 90? (14) Find 5% of 62. (15) 33 13 % of 24 is what? Solution Intervention 4-1 (A) 0.09;

(B) 0.67;

(G) 0.00006;

(C) 0.002;

(H) 1;

(I) 0.5;

1 1 1 · = ; (F) 0.0945; 2 100 200 10 1 1 (J) · = ; (K) 0.067; (L) 0.01 3 100 30 (D) 6;

(E)

Solution Intervention 4-2 (1) 12%; (2) 40%; (3) 4.5%; (4) 290%; (5) 600%; (6) 1.03%; (7) 3000%; (8) 1490%; (9) 20%; (10) 75%; (11) 100%; (12) 9%; (13) 10000%; (14) 100.9%; (15) 0.02% Solution Intervention 4-3 M 9

E 7

G 5

A 4

H 6

E 3

R 2

T 1

Z 8

A Megahertz, abbreviated MHZ, is the speed of microprocessors, representing one million cycles per second. 1 100 1 1 (4) 33 % = · = 3 3 100 3 5 50 1 1 5 %= · = 9 9 100 18 2 200 1 2 66 % = · = 3 3 100 3 Solution Intervention 4-4 T 9

O 2

N 6

Y 8

F 10

A 5

D 7

E 1

L 4

L 3

In 2001, the iPod originated with a business idea dreamed up by Tony Fadell. 109

CHAPTER 9

Solution Intervention 4-5

Solution Intervention 4-6 P 6

A 2

S 5

C 4

A 3

L 1

Blaise Pascal (1623–1662) was a French mathematician who invented the first calculating device. Solution Unit #4 Supplemental Quizzes 1 Quiz 4A: (a) 0.45; (b) 50%; (c) ; (d) 75%; (e) 10. 2 1 Quiz 4B: (a) 0.05; (b) 75%; (c) ; (d) 50%; (e) 1.5. 4 1 Quiz 4C: (a) 0.68; (b) 85%; (c) ; (d) 20%; (e) 9. 5 1 Quiz 4D: (a) 1.75; (b) 350%; (c) ; (d) 25%; (e) 3. 2 110

SPECIFIC CONTENT AREA INTERVENTIONS – PERCENTS (UNIT IV)

Solution RAR – Post Unit Assessment (4) (1) 0.35;

(2) 0.09;

(7) 79%;

(8) 320 %;

(12) 85%;

(13) 36;

(3) 0.057;

(4) 0.008;

3 1 (9) ; (10) ; 5 20 (14) 3.1; (15) 8

(5) 2.09;

(6) 40%;

(11) 80%;

GRADING SCALE RAR – POST UNIT ASSESSMENT (4)

111

CHAPTER 10 SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY (UNIT V)

This chapter addresses some basic number concepts (writing number in words, ordering, rounding, averages, order of operations, and proportions) and geometry (finding the area and perimeter of quadrilaterals and triangles) that all students should have a general understanding of before taking algebra. Here are tips and strategies that might be helpful with certain concepts: − If students are having difficulty writing decimal numerals in words, it might be helpful to have them first read the decimal part as if it were a whole number followed by naming the place value of the last digit. Here is an example: For 2.0349, the decimal part is three-hundred forty-nine and the last digit (9) is in the ten-thousandths column. Putting it all together, we have two and three-hundred forty-nine ten-thousandths. − For difficulty in ordering decimals, have students add enough zeros to the end so that all numbers have the same number of digits then compare as if whole numbers. Let us compare 0.18, 0.2, 0.109, and 0.08 as an example. Since the most digits to the right of the decimal in any of the numbers are three, let’s add enough zeros so that each number has three digits to the right of the decimal, giving us: 0.180, 0.200, 0.109, 0.080. Now just ignore the decimals and compare as if all were whole numbers, such as 80, 109, 180, and 200. Therefore, in order from least to greatest, we have: 0.08, 0.109, 0.18, and 0.2. − For rounding keep it simple: Have students locate the place value that the number is to be rounded, then check the number to the right; if it is five or more (5, 6, 7, 8, 9) round the identified place value up and if less than 5 (0, 1, 3, 4) leave it alone. In either case, for whole numbers fill in the place values to the right with zeros and for decimals “cut off” the numbers after the identified place value. For example, in rounding 6.0487 to the nearest hundredth, the “4” is in the hundredths column and to its right there is an “8” which is five or more. Therefore, we round the hundredth column up, leaving us with 6.05.

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CHAPTER 10

− Use finding the average to improve your students’ addition and division skills with whole numbers, fractions, and decimals. − For order of operations, the acronym PEMDAS (for Parentheses, Exponents, Multiplication/Division (whichever occurs first from left to right), Addition/Subtraction (whichever occurs first from left to right) might be helpful. Mnemonically it is also expressed as “Please Excuse My Dear Aunt Sally”. − In solving proportions, have students multiply the two numbers that are diagonally across from each other then divide the product by the other number. For example, in n6 = 34 , multiply 6 × 4 then divide by 3, which is 8. − At a minimum, students in preparation for algebra should be able to find the area and perimeter of a square, rectangle, and triangle. Perimeter should be fairly easy, but please remind students to add the lengths of all sides. Remind students that area is measured in square units and have them memorize the basic formulas (square = s2 , rectangle = l × w; and triangle = 12 bh).

114

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Intervention #5-1

WRITING NUMBERS IN WORDS (5-1)

Keep in mind that you write decimals just as you would read decimals. Here are some steps to make the process easier: Take for example, 12.5398. 1. Read the whole number – twelve. 2. Substitute the word “and” for the decimal point – twelve and . . . 3. Read the decimal part as if it were a whole number: 5,398 (use commas if you need help) – five thousand three-hundred ninetyeight. 4. Name the place value of the last digit. The eight is in ten-thousandth place. Thus, twelve and five thousand three-hundred ninety-eight ten thousandths. Directions Write the following numbers in words: (1) 12.05 (2) 7.108 (3) 0.0008 (4) 4,398.1 (5) 18.09001 (6) 0.00902 (7) 7,091,632 (8) 550.550 (9) 5.012345 (10) 10,0000.0001

115

CHAPTER 10

Intervention #5-2

RECOGNIZING NUMBERS OF GREATER AND LESSER VALUE (5-2)

When comparing decimals, one method is to place enough zeros at the end of each of the numbers so that the numbers have the same number of digits to the right of the decimal. Once you do that, remove the decimals and compare the numbers as if they were whole numbers. Let us use 0.809 and 0.81 as an example. If both numbers had three digits to the right of the decimal, we would be comparing 809 and 810, thus 0.81 is the greatest number. For fractions, you should rewrite each fraction using a common denominator (LCM) then compare numerators. Directions Place the correct symbol <, >, or =. (1) 2.07

2.1

(2) 0.25

0.206

(3) 0.1304

0.13038

(4) 0.705 5 8 7 (6) 9 (5)

0.7050 7 12 21 27

(7) 0.09076 (8)

4 15

0.091909 7 20

(9) 0.601 (10) 10.90008

116

3 5 10.9807

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Intervention #5-3

RECOGNIZING NUMBERS OF GREATER AND LESSER VALUE (5-2)

Directions Chose two of the six choices below, which are in order from least to greatest. (A) 0.2, 0.04, 0.51, 6.78 (B) 0.035, 0.1, 4.5, 4.45 (C) 0.07, 0.17, 0.707, 0.71 (D) 0.903, 3.08, 3.108, 3.02 (E) 0.503, 0.9, 0.918, 0.92 1 3 (F) 0.45, , 0.51, , 0.081 2 4

117

CHAPTER 10

Intervention #5-4

ROUNDING DECIMALS AND WHOLE NUMBERS (5-3)

When rounding, keep in mind the following: 1. Locate the place value to which you are rounding. 2. Evaluate the place value to the right. If that place value is 5 or more (5, 6, 7, 8, or 9) round the given place value up one. If that place value is less than 5 (0, 1, 2, 3, or 4), leave the given valueas it is and then remove all digits to its right. Directions Round each number to the given place value. (1) Round 8,097 to the nearest hundred. (2) Round 19.71 to the nearest tenth. (3) Round 8.0987 to the nearest thousandth. (4) Round 5,600.0905 to the nearest thousand. (5) Round 0.09083 to the nearest ten-thousandth. (6) Round 0.6590 to the nearest one (unit). (7) Round 578.5064 to the nearest hundredth. (8) Round 9.983754 to the nearest hundred-thousandth. (9) Round 7.85 to the nearest ten. (10) Round $76.449 to the nearest dollar. (11) Round 15.098 to the nearest tenth. (12) Round $128.995 to the nearest cent.

118

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Intervention #5-5

FINDING THE AVERAGE (5-4)

The average or mean of a group of numbers is determined by getting the sum of a group together then dividing by the number of numbers in the group. For example, in the group 22, 24, 11, 7, 32, 24, the sum is 120. Dividing by 6 we get an average of 20. Directions Find the average of each group of numbers. (1) 43, 52, 15, 70, 85 (2) 24, 36, 88, 72 (3) 321, 450, 873 (4) 2.4, 0.8, 16, 5.2 (5) 27, 49, 80, 63, 78, 21 1 2 3 (6) , , 2 3 4

119

CHAPTER 10

Intervention #5-6 ORDER OF OPERATIONS (5-5)

In order to have uniform answers that are correct, we must follow a certain order when working math problems. The order of operations is as follows: 1. 2. 3. 4.

Perform the operations inside the parentheses first. Simplify your exponents. Multiply or divide depending on which occurs first from left to right. Add or subtract depending on which occurs first from left to right.

For example: (3 + 2)2 + 9 ÷ 3 First, simplify inside your parentheses (3 + 2 = 5). Second, simplify the exponent (52 = 25). Third, divide (9 ÷ 3 = 3). Last, add (25 + 3 = 28). Directions Simplify the following problems, then match your answers with the choices on the right to answer the following question: When all know that Neil Armstrong was the first man to walk on the moon, however, who was the second? (1) 12 + 8 · 3

A. 28

(2) 15 ÷ 3 + 6 · 2

B. 2

(3) (8 + 4) ÷ 2 − 5

D. 20

(4) 12 + 32 · 4

I. 17

(5) (6 − 4) + 3 ÷ 3

L. 0

(6) 42 − 2 ÷ 2 + 5

N. 3

(7) 6 · (5 + 3) ÷ 22

R. 36

(8) 6 + 12 ÷ 3 − 2 · 5

U. 12

(9) 4 ÷ 22 × (5 − 3)

Z. 48

(10) (3 + 2)2 + 9 ÷ 3

Z. 1

9 120

7

3

4

10

8

6

1

2

5

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Intervention #5-7

SOLVING PROPORTIONS (5-6)

A proportion is an equation with a ratio or fraction on each side. A short-cut to solving proportions is to multiply the two numbers that are diagonally across from each other then divide by the other number. Directions Solve the proportions below then match your answers to the choices on the right to answer the following question: Who invented the first artificial heart? (1)

(4)

(7)

3 9 = 4 n

(2)

18 9 = 4 n

(5)

n 9 = 8 3

(10) 11

(8)

n 3 = 1 2 5 1

8 n = 4 5

3 12 = n 4

3

(6)

(9)

n 2.5 = 6 3 6

n 18 = 1 3

A. 10 B. 1 E. 6 1 I. 1 2 1 J. 2

7 n = 4 8

K. 30 O. 12

n 15 = 2 20

(11) 5

(3)

7

2 7 = n 28

(12) 12

3 4

n

= 2

R. 14 R. 2 R. 5 T. 24

4 3 8 9

V. 8 4

9

8

10

121

CHAPTER 10

Intervention #5-8 FINDING THE PERIMETER OF A SQUARE AND RECTANGULAR (5-7)

Remember, that the perimeter of a polygon (a closed figure made-up of line segments e.g. triangle, square, rectangle, etc.) is the sum of the lengths of all its sides. Keep in mind that a rectangle as two equal lengths and two equal widths and that a square has four sides of equal length and width. Directions Find the perimeter of each figure given its dimensions and then use your answers to complete the BINGO table below. There is only one way to win, so good luck. (Show all work and no calculators please!) (1)

(2) 7 cm 11 m

(3) Rectangle: width 4 cm; length 8 cm. (4) Square: length 5 cm. (5) Rectangle: length 19 cm; width 3 cm. (6) Square: width

1 4

cm.

(7) Triangle: sides of 7 cm, 8 cm, and 12 cm. (8) Rectangle: width 3.5 cm; and length 7 cm. (9) Square: length: 4.6 cm. (10) Rectangle: length 5 23 cm; width 8 34 cm.

122

8 cm

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Intervention #5-9

FINDING THE AREA OF A SQUARE AND RECTANGULAR (5-7)

Area is measured in square units. The area of a rectangle is determined by multiplying its length times it width. The formula for the area of a square is s2 , which simply means squaring the length of its side. Directions Find the areas of the following polygons and then cross out the boxes with the correct answers below, leaving you with the letters that spell a “key” word for your success. (1)

(2) 4 cm 8 cm 6 cm

(3) Rectangle: 9 cm length; 5 cm width. (4) Square: 5 cm length. (5) Rectangle: 3 cm width; 13 cm length. (6) Square: 4 cm width (remember in a square that the length and width are equal). (7) Rectangle: 6 cm length; width

2 3

cm.

(8) Triangle: base 10 cm; height 9 cm (hint A = 1/2 bh). (9) Square: length: 5 12 cm. (10) Rectangle: 4.75 cm width; 1.2 cm length.

123

CHAPTER 10

124

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY RAR – POST TEST (5) NUMBER CONCEPTS AND BASIC GEOMETRY

Directions Solve to the following problems, showing as much work as needed. Calculators are not allowed for this assessment. (5.1) Writing numbers in words (1) Write 2,097,820.5 in words:

(2) Write 9.0802 in words:

(3) Write six-hundred eleven and twenty-nine thousandths in standard form:

(5.2) List in order from least to greatest (4) 6.4,

0.76,

0.8,

0.093

(5) 0.19,

0.3,

2.4,

0.039

5 (6) , 8

2 , 5

3 , 4

1 2

(5.3) Rounding (7) Round 90,534.7873 to the nearest thousand. (8) Round 7.0941 to the nearest thousandth. (5.4) Find the average of the following group of numbers (Show all work!) (9) 225,

87,

63

(10) 33,

82,

25,

(11) 8.2,

9.7,

17,

115,

95

13.1

(5.5) Compute the following: (12) 18 − 9 ÷ 3 + 2

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CHAPTER 10

(13) 16 + 35 ÷ (6 − 1) − 32 (5.6) Solve for n in the following proportions: (14)

9 3 = n 2

(15)

9 n = 1 5

(16)

9 3 = 21 n

(5.7) and (5.8) Given the rectangle and square, solve the problems below: (A)

(B) 4 cm

5 cm

7 cm 5 cm (5.7) Answer the questions below: (17) What is the perimeter of Figure A? (18) What is the perimeter of Figure B? (5.8) Answer the questions below: (19) What is the area of Figure A? (20) What is the area of Figure B? Solution Intervention 5-1 (1) 12.05 = twelve and five hundredths (2) 7.108 = seven and one hundred eight thousandths (3) 0.0008 = eight ten-thousandths (4) 4,398.1 = four thousand three hundred ninety-eight and one tenths (5) 18.09001 = eighteen and nine thousand one hundred-thousandths (6) 0.00902 = nine hundred two hundred-thousandths (7) 7,091,632 = seven million, ninety-one thousand, six hundred thirty-two (8) 550.550 = five hundred fifty-five and fifty-five hundredths (9) 5.012345 = five and twelve thousand three hundred forty-five millionths (10) 10,000.0001 = ten thousand and one ten-thousandth Solution Intervention (5-2) (1) 2.07 < 2.1 (2) 0.25 > 0.206 (3) 0.1304 > 0.13038 (4) 0.705 = 0.7050 126

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

5 7 15 14 > = > 8 12 24 24 7 21 21 21 (6) = = = 9 27 27 27 (7) 0.09076 < 0.091909 (5)

4 7 16 21 < = < 15 20 60 60 3 (9) 0.601 > = 0.601 > 0.6 5 (10) 10.90008 < 10.9807 (8)

Solution Intervention (5-3) C and E (A) 0.20, 0.04, 0.51, 6.78 (B) 0.035, 0.100, 4.500, 4.450 (C) 0.070, 0.170, 0.707, 0.710 (D) 0.903, 3.080, 3.108, 3.020 (E) 0.503, 0.900, 0.918, 0.920 1 3 (F) 0.45, , 0.51, , 0.081 2 4 Solution Intervention (5-4) (1) 8,100;

(2) 19.7;

(7) 578.51;

(3) 8.099;

(8) 9.98375;

(9) 8;

(4) 6,000;

(5) 0.0908;

(10) $76;

(11) 15.1;

(6) 1; (12) $129

Solution Intervention (5-5) (1) 53;

(2) 55 ;

(3) 548;

(4) 6.1;

(5) 53;

(6)

23 36

Solution Intervention (5-6) B 9

U 7

Z 3

Z 4

A 10

L 8

D 6

R 1

I 2

N 5

Colonel Edwin (Buzz) Aldrin Jr. landed on the moon at the same time as Neil Armstrong, the first man to actually walk on the moon. However, Aldrin was the second man to walk on the moon. 127

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Solution Intervention (5-7) R 11

O 1

B 5

E 3

R 6

T 7

J 12

A 2

R 4

V 9

I 8

K 10

In 1978, Robert Jarvik invented the first artificial heart. Solution Intervention (5-8) (1) 36 cm;

(2) 32 cm;

(3) 24 cm;

(7) 27 cm;

(8) 21 cm;

(9) 18.4 cm;

(4) 20 cm; (10) 28

(5) 44 cm;

(6) 1 cm;

5 6

Solution Intervention 5-9 “MATH” is a key to success. (1) 32 cm2 ;

(2) 36 cm2 ;

(6) 16 cm2 ;

(7) 4 cm2 ;

(3) 45 cm2 ; (8) 45 cm2 ;

(4) 25 cm2 ; (9) 30 14 cm2 ;

(5) 39 cm2 ; (10) 5.7 cm2

Solution Unit #5 Supplemental Quizzes Quiz 5A: (a) twelve and seventy-three thousandths; (b) 1, 4, 2, 3 (c) 4.35; (d) 101; (e) 45. Quiz 5B: (a) 12; (b) 24 cm; (c) 32 cm2 ; (d) seven and nine hundred eighty-four ten-thousandths. 128

SPECIFIC CONTENT AREA INTERVENTIONS – NUMBER CONCEPTS AND BASIC GEOMETRY

Quiz 5C: (a) 2, 4, 3, 1; (b) 4.09; (c) 54; (d) 1; (e) 1. Quiz 5D: (a) 36; (b) 81 cm2 ; (c) two million ninety-eight thousand one hundred eighty; (d) 3, 2, 1, 4; (e) 5.988. Solution Unit 5 Post Assessment (1) Two million ninety-seven thousand eight hundred twenty and five tenths (2) Nine and eight-hundred two ten-thousandths (3) 611.029;

(4) 0.093, 0.76, 0.8, 6.4;

2 1 5 3 (6) , , , ; (7) 91,000; 5 2 8 4 (11) 12; (12) 17; (13) 14; (17) 22 cm;

(18) 20 cm;

(8) 7.094; (14) 6;

(19) 28 cm2 ;

(5) 0.039, 0.19, 0.3, 2.4; (9) 125; (15) 45;

(10) 70; (16) 7;

(20) 25 cm2

GRADING SCALE UNIT 5 POST ASSESSMENT

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CHAPTER 11 SPECIFIC CONTENT AREA INTERVENTIONS – SOLVING BASIC WORD PROBLEMS

Students must be able to solve basic word problems, determining which operation (addition, subtract, multiplication, division) or combination of two operations are needed to solve a problem. In reviewing strategies with your students for solving basic word problems, stress to them that certain terms/words may indicate specific mathematical operation. The following is a partial list:

Addition

increased by sum added to more than combined, together total of

Subtraction

difference decreased by minus, less difference between/of less than, fewer than

Multiplication

of times, multiplied by product of

Division

per, a out of ratio of, quotient of percent (divide by 100)

Equals

is, are, was, were, will be same as gives, yields sold for

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Here are some additional tips and strategies that will help students struggling with basic word problems: − Encourage students to read problems carefully looking for operational terms/phrases, cues, and important information. (Emphasize that reading a problem more than once is not a weakness, but smart.) − As they read, encourage student to sort through various information to determine exactly what it is being asked. Encourage students to write down important information (numbers, operational terms/phrases, etc). − Where appropriate, have students draw pictures and/or diagrams to improve their understanding and to determine how to solve the problem. − Encourage students to develop a clear/confident understanding (based on the information) of the operation(s) used to find the solution. − Have students check their answers for reasonableness. (Does the answer make sense?)

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Intervention 6-1

SOLVING WORD PROBLEMS (6-1)

Directions Find the solutions to the word problems below and then match your answers with the choices to the right to answer the following question: Who was the Greek Mathematician who founded geometry? (No calculators please!) (1) Jamel purchased a CD for $14.75, a cap for $11.30, and a pair of socks for $4.25. What was the total cost of the items?

A. $52.75 B. 845 ft

(2) If Keisha’s total purchase at the GAP came to $18.65 and she gave the sales clerk a $20 bill, how much change she should she get back?

C. $1.35 D. $32.40

(3) Hillary swam 1.5 miles on Tuesday, 1.75 miles on Thursday, and 2 miles on Saturday. How many miles did she swim on all three days?

E. 2,055 ft H. $29.30

(4) For a local charity, Cheryl collected $55.70 and Perry collected $88.10. How much more did Perry collect than Cheryl?

I. $30.30 K. $2.35

(5) Kareem received a $7.50 discount off the purchase milesprice of a school uniform. If the original price of the uniform was $45.25, what was its cost after the discount?

L. 5.25 U. $37.75

(6) Seattle’s Space Needle stands 605 feet tall and Willis Tower, formerly named Sears Tower, in Chicago stands about 1,450 feet. If you could put the Space Needle on top of the Willis Tower, how high they reach?

6

5

2

3

1

Y. 4.2 miles Z. $143.50

4

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Intervention 6-2

SOLVING WORD PROBLEMS (6-2)

Directions Find the solutions to the word problems below then cross out the boxes with the correct answers below, leaving you with the letters that spell another “key” word for your success. (No calculators please!) (1) At her favorite clothing store, Deanna bought four pair of jeans costing $19 each. What was her total cost for all four pairs of jeans? (2) Grady works part-time at the local movie theater. Last week he worked 15 hours and earned $90. How much did Grady earn for each hour he worked? (3) Tracy baby sat for Aunt Maria and earned $6.50 per hour. If she works 5 hours, how much will she earn? (4) The Garcia’s family dinner at their favorite restaurant came to a total of $86. If they want to leave the waiter a 15% tip, how much tip should they leave him? (5) Albert consumes about 13,370 calories per week. About how many calories does he eat each day? (6) Eight good friends ate dinner that the mall’s Pizza Parlor and the bill was to be share equally. If the total bill was $60.80, how much does each person owe? (7) Jacob received a 20% discount on a skateboard that normally costs $48. How much did he save on the skateboard? (8) Deshawn borrowed $180 from his dad interest free and agreed to pay back the money in equal monthly payments over the next year. How much should he pay back each month?

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Intervention 6-3

SOLVING WORD PROBLEMS (6-3)

Directions Find the solutions to the word problems below and then match your answers with the choices to the right to answer the following question: Who was the physical education instructor who invented basketball? (No calculators please!) (1) Andy bowled three of games with scores of 162, 150, and 135. What was his average score?

A. 27 B. 1

(2) Renee earns $8.00 an hour working at local hardware store. Because of her outstanding work, her boss plans to give her a 20% raise. What will her new salary be including the raise?

E. 25 hours H. 21

(3) Eli wants to buy a pair of football cleats that cost $50. In his State, there is a 6% sales tax. What will be the total cost of the shoes including the sales tax?

I. 53 I. 29

(4) Payton wants to bike 78 miles over 3 days. He bikes 22 miles on the on the first day and 29 miles on the second. How many miles must he bike on the third day to reach his goal?

M. 149 N. 3.0

(5) Each student at Centreville Middle School is required to complete miles 30 hours of community service. So far, Marcus has performed 22.5 hours and his sister Cheri has three hours more than he. How many more hours of community service does Cheri need to perform?

O. 122 S. 9.6 T. 4.5

(6) Brandon wants to a buy a video game for $20. If there is 5% sales tax in his State, what is the total cost of the game including tax?

U. 4 V. 5

(7) Jenna is to receive a 10% discount off a blouse that original costs $30. What is the price of the blouse after the discount? (8) Christopher is trying to determine his grade point average (GPA) for for this quarter. If he had six classes this quarter with following grades: A (4), B (3), A (4), C (2), B (3), C (2); what was his GPA? 8

7

4

2

1

3

5

W. 1.6 Y. 447

6

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CHAPTER 11

Intervention 6-4

FUN WITH WORD PROBLEMS (6-1, 6-2, 6-3)

Directions Find the solutions to the following word problems. Be careful, in addition to being fun, some of these can be tricky. (1) What is 10 divided by

1 2

then added to three?

(2) When four friends get together, each friend shakes hands with each of the other friends. How many handshakes will there be all together? (3) Sara is twice as old as Sam. Stan is twice as old as Sara. The sum of their ages is 28. How old are Sara, Sam, and Stan? (4) Alan, Lori, Jabar, and Macy all participate in four different sports. Their sports are track, golf, tennis, and basketball. Lori is the sister of the tennis player. Macy’s sport does not use a ball. Jabar once made a birdie in his sport. Determine which sport each plays. (5) At the end of a school race, Jackie is 15 meters behind Wally. Wally is 5 meters ahead of Brandon. Brandon is 15 meters ahead of Peter. Peter is 5 meters behind Jackie. In what order did they finish the race? (6) The Garcia Family (Mr. Garcia, Mrs. Garcia, their son Jorge, and daughter Maria) live across a canyon and the only way home is by way of a hot air balloon. Someone must always be in the balloon to control it. The balloon can carry only 200 pounds at one time. Mr. Garcia weighs 150 pounds, Mrs. Garcia weighs 125 pounds, Jorge and Maria each way 100 pounds. How can we get the Garcia family across the canyon to their home? (7) Find the number that will logically continue each of the sequences: (A) 1, 1, 2, 3, 5, 8, 13, (B) 11, 13, 17, 25, 32, 37, (8) As a member of her school’s track team, Keisha is training for the 100-meter dash. Her practice times (in seconds) this week were13.2, 13.75, 13.1, 14.15, and 13.85. What was Keisha’s average time for the 100-meter dash this week?

136

SPECIFIC CONTENT AREA INTERVENTIONS – SOLVING BASIC WORD PROBLEMS

(9) Which is the better buy? (A) 28 oz jar of spaghetti sauce that costs $4.20 or (B) 16 oz jar of spaghetti sauce that costs $3.20 (10) Derrick is hoping to make the honor roll. At his school, Derrick needs a 3.0 grade point average to make the honor roll. Derrick’s report card is as follows: Science, A; French, C; English, B; P.E., A; Social Studies, D; Math, A; and Band, B. Based on a four-point scale (A=4, B=3, C=2, D=11, F=), does Derrick make the honor roll?

137

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138

SPECIFIC CONTENT AREA INTERVENTIONS – SOLVING BASIC WORD PROBLEMS RAR – UNIT TEST (6) WORD PROBLEMS

Directions Solve to the following word problems, showing as much work as needed. Calculators are not allowed for this assessment. (1) The Maxwell family drove 589 miles on Thursday, 423 miles on Friday, and 503 miles on Saturday. How many miles did they drive on all three days? (6.1) (2) If Vernon purchased a CD for $12.76 at the local music store and gives the sales clerk a $20 bill, how much change should he get back. (6.1) (3) Mount Everest stands 29,028 feet and Mount Kilimanjaro stands 19,300. How much higher is Mount Everest than Mount Kilimanjaro? (6.1) (4) The Willis Tower in Chicago stands 1,450 feet and the Washington Monument is about 555 feet tall. If you could put the Willis tower on top of the Washington Monument, how high would the structures reach? (6.1) (5) This weekend at the mall, Bruce bought three shirts costing $16 each. What was the total cost on all three shirts? (6.2) (6) Alexis works part-time at the local pet store. Last week she worked 14 hours and earned $112. How much did Alexis earn for each hour of she worked? (6.2) (7) Christopher reads about 462 pages each week. How many pages does he read each day? (6.2) (8) Leah baby sits for her neighbor and earns $7.50 per hour. If she works 5 hours, how much will she earn? (6.2) (9) Playing three rounds of golf, Anderson had scores of 81, 75, and 72. What was his average score for the three rounds of golf? (10) Joann received a 33 13 % discount on the purchase of a dress that originally cost $60. What is the cost of the dress after the discount? (11) Jack wants to sail 155 miles over 3 days. He sails 43 miles on the first day and 58 miles on the second. How many miles must he sail on the third day to reach his goal? (6.3) (12) Each student at Excellence Middle School is required to complete 35 hours of community service each year. So far, Christina has performed 27 hours and her best friend, Jean, has performed 3 hours more than she. How many more 139

CHAPTER 11

hours of community service does Jean need to perform to meet the school’s requirement? (6.3) Solution Intervention 6-1 E 6

U 5

C 2

L 3

I 1

D 4

Euclid (330–275 BC), also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the “Father of Geometry”. Solution Intervention 6-2 “ALGEBRA” Research shows that “algebra” is a major key to students’ success in college and the workplace. (1) $76; (2) $6; $ 9.60; (8) $15

(3) $32.50;

(4) $12.90;

(5) 1910 cal;

(6) $7.60;

(7)

Solution Intervention 6-3 N 8

A 7

I 4

S 2

M 1

I 3

T 5

H 6

James Naismith (1861–1939) was a Canadian physical education instructor who invented basketball in 1891. Solution Intervention 6-4 (1) 10 ÷

1 2

= 10 × 2/1 = 20 + 3 = 23

(2) Six handshakes (3) Sara is 8; Sam is 4; and Stan is 16 (4) Jabar is the golfer, Macy runs track, Lori plays basketball, and Alan play tennis (5) Wally, Brandon, Jackie, and Peter (6) Solution: 1. The two kids, Jorge and Maria, take the balloon across the canyon. 2. One kid stays and the other returns the balloon. 3. One of the parents then takes the balloon home across the canyon. 140

SPECIFIC CONTENT AREA INTERVENTIONS – SOLVING BASIC WORD PROBLEMS

4. The parent stays home while the kid, who was home, returns the balloon across the canyon. 5. Next, both kids take the balloon home across the canyon. 6. One kid stays and the other returns the balloon. 7. Next, the second parent takes the balloon home across the canyon. 8. The second parent stays home while the kid, who was home, returns the balloon across the canyon. 9. Finally, both kids take the balloon home. (7) (a) 1, 1, 2, 3, 5, 8, 13, 21. Pattern: Add the number that precedes the given number. For example, 2 + 1 = 3; 3 + 2 = 5; 5 + 3 = 8; 8 + 5 = 13, etc. (7) (b) 11, 13, 17, 25, 32, 37, 47. Pattern: Add the sum of the digits to the given number to get the next number. For example, 11 + (1 + 1) = 13; 13 + (1 + 3) = 17; 17 + (1 + 7) = 25; 25 + (2 + 5) = 32, etc. (8) 68.05 ÷ 5 = 13.61 28 16 = 0.15 cent; = 0.20 cent. $4.20 3.20 (10) Yes, (4 + 2 + 3 + 4 + 1 + 4 + 3) = 21 ÷ 7 = 3.0 (9) A,

Solution Unit #6 Supplemental Quizzes Quiz 6A: (a) 1,275; (b) 8; (c) $6.75. Quiz 6B: (a) $11; (b) $7.61; (c) 188. Quiz 6C: (a) 5.25; (b) $54; (c) $21.20. Quiz 6D: (a) $41.25; (b) $10; (c) $12 per hour. Solution Post Unit Assessment (6) (1) 1515 miles; hour; (7) 66;

(2) $7.24; (3) 9,728 ft; (4) 2005 ft; (5) $48; (6) $8 per (8) $37.50; (9) 76; (10) 40; (11) 54 miles; (12) 5 hours.

GRADING SCALE UNIT 6 POST ASSESSMENT

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CHAPTER 12 SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

It is quite advantageous for students going into algebra to be skilled in performing operations (adding, subtracting, multiplying, and dividing) with rational numbers (negative and positive whole numbers, fractions, and decimals). Also, students should be able to solve basic one – and two-step equations. In reviewing these prealgebraic concepts with your students, remember to keep the processes as simple as possible. For example, subtraction is simply adding the inverse of the number being subtracted. Therefore, once students have mastered adding rational numbers, subtraction should be easy. Here are a few process pointers that you might use to simplify addition for your students. − When adding numbers with like signs, add the absolute values and keep the same sign. − When adding numbers with unlike signs, get the difference of the absolute values and then use the sign of the number with the greater absolute value. From students that have difficulty with subtracting, it is recommended that they rewrite the subtraction problems using addition, adding the inverse of the number being subtracted. As you know, the process for multiplying and dividing signed numbers is even easier: like signs results in a positive product or quotient and unlike signs results in a negative product or quotient. Finally, for solving basic one- and two-step equations, it is just as important for students to learn the proper technique (horizontal and/or vertical) as it is obtaining the correct answer. The proper technique and logical reasoning skills used to solve these simple equations will enable students to later solve more complex multi-step equations that they will have in algebra. As you know, in solving these basic equations the objective is to isolate the variable on side of the equation or the other. And this is done by performing the opposite operation(s) on both sides of the equation to eliminate the other terms. Here are a few examples: (A)

x + 3 = −5 x + 3 − 3 = −5 − 3 x = −8

143

CHAPTER 12

(B) −3x = 12 −3x = 12 −3 −3 x = −4 (C)

144

5x + 4 5x + 4 − 4 5x 5 x

= 19 = 19 − 4 = 15 5 = 3

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

Intervention #7-1

ADDING INTEGERS (7-1)

When adding like signs, positive plus positive or negative plus negative, simply add the absolute values then keep the sign for your sum. Examples: (A)

5 + 9 = 14

(B)

− 5 + (−9) = −14

Directions Find the sum of the following problems and then match your answers with those below and place the corresponding letter above it to complete the statement. (A) 4 + 13 = (C) −4 + 13 = (T) −20 + 11 = (S) −21 + 31 = (T) 3.5 + 4.5 = (I) 19 + −24 = (S) 18 + −15 = 1 3 + = 4 4 (T) −35 + 34 = (I)

(S) −37 + 37 =

0

−1

17

8

−5

3

−9

1

9

10

includes data collection, analysis, and interpretation.

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CHAPTER 12

Intervention #7-2

ADDING INTEGERS (7-1)

When adding unlike signs, subtract the absolute values of the numbers then use the sign of the number with greatest absolute value in your answer. Examples: (A)

− 9 + 16 = 7

(B)

9 + (−16) = −7

Directions Find the sum of the following problems and then match your answers with those below and place the corresponding letter above it to complete the statement. (U) −8 + 6 =

(U) 33 + (−18) =

(C) −9 + 11 =

(I) 37 + (−50) =

(T) 12 + (−15) =

(S) − 15 + (−15) =

(L) 13 + (−9) =

(S) 15 + 15 =

(A) 6.3 + (−0.3) =

(U) 0 + (−28) =

1 1 (N) −1 + = 4 4 (L) −27 + 10 =

(F) −2.5 + 2.5 =

(N) −21 + 29 =

(C) −19 + 6 + (−8) =

(C) −11 + 5 + 11 =

(O) 39 + (−46) =

−21

6

−17

5

−2

4

15

30

is a higher mathematics that deals with differentiation and integration of 0

146

−28

−1

2

−3

−13

−7

8

−30

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

Intervention #7-3

SUBTRACTING INTEGERS (7-2)

Subtracting is the same as adding the opposite (or additive inverse) of the number being subtracted. Examples: (A)

− 5 − 10 = −5 + (−10) = −15

(B)

12 − (−23) = 12 + 23 = 25

Directions Find the difference in each of the following problems and then cross out the boxes of the correct answers below, leaving you with some letters that you must unscramble to make “two words”. (1) −6 − 12 =

(11) −9 − 20 =

(2) 13 − 23 =

(12) 23 − 40 =

(3) −17 − (−20) =

(13) −19 − (−25) =

(4) −24 − (−16) =

(14) −26 − (−3) =

(5) 18 − (−20) =

(15) 29 − (−29) =

(6) 0 − 19 =

(16) 0 − (−21) =

(7) −34 − 12 =

(17) −18 − 20 =

(8) 15 − 40 =

(18) 16 − 32 =

(9) −32 − (−44) =

(19) −2.5 − (−3.9) =

(1) −71 − (−49) =

7  1 (20) − − − = 8 2

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CHAPTER 12

Intervention #7-4

MULTIPLYING AND DIVIDING INTEGERS (7-3 AND 7-4)

Multiplication and Division Made Easy: When multiplying or dividing rational numbers (negative and positive whole numbers, fractions, decimals, etc.) the rules are easy to remember. When multiplying or dividing two unlike signs (− + or + −) the product or quotient is always negative. When multiplying or dividing like signs (+ + or − − ) the product or quotient is always positive. Here are a few examples: (A)

− 5 · 3 = −15

(C)

18 ÷ −6 = −3

(B)

− 15 ÷ −3 = 5

(D)

− 7 · −8 = 56

Directions Compute the problems below and then match your answers with the choices to the right to answer to the following question: Who invented the first Microwave Oven? (No calculators please!) (1) −28 ÷ −7 =

C. 0

(2) −12 · 4 =

C. −4

(3) 2.3 · −1.4 =

E. 90

(4) 111 ÷ −3 =

E. 4

24 = −6 7 8 (6) − · − 8 11 (7) (−2)3

E. −37

(8) −23 · 0 =

P. −3

(9) −120 ÷ −6 =

R. −3.22

(10) −3 · 5 · −6 =

R. 20

(11) −2.5 ÷ 0.5 =

S. −5

(12) −3 + 5 · (−2) =

Y. −8

(5)

12

148

1

7 11 P. −13

N.

3

8

7

11

2

4

6

5

10

9

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

Intervention #7-5

OPERATIONS WITH INTEGERS (7-1, 7-2, 7-3, 7-4) BINGO!

Directions Cross out the correct answers as you compute the problems below. If done correctly, everyone should be winner (horizontally, vertically, or diagonally). Find the winning block of numbers. (1) −19 + (−11)

(9) 4(−8)

(2) −8 + 17

(10) (−5)(−3)

(3) −16 + 9

(11) (−4)2

(4) 19 + (−18)

(12) (−2)(5)(−1)

(5) 11 − (−20)

(13)

(6) 9 − 21

(14) −18 ÷ −6

(7) −18 − (−4)

(15)

48 −4 −44 4

(8) −11 − 14

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Intervention #7-6

SOLVING EQUATIONS (7-5)

Keep in mind, that in order to solve basic equations, you must cancel terms by performing the opposite operation to both sides of the equation. Division and multiplication are opposite operations. Addition and subtraction are opposite operations. Directions Identify the correct procedures for solving the following problems: Example: x − 8 = 15 To solve, add 8 to both sides of the equation. (1) x + 7 = 9

(a) divide by 7

(2) x − 7 = 9

(b) subtract 9

(3) 7x = 63 x (4) = 9 7 (5) 7x − 9 = 40

(c) multiply by 7 (d) subtract 7 then divide by 9

(6) 7 = x − 9

(f) subtract 7

(7) 7 = x + 9

(g) add 9 then divide by 7

(8) 63 = 9x x (9) 4 = + 7 9 (10) 25 = 9x + 7

(h) subtract 7 then multiply by 9

150

(e) add 9

(i) divide by 9 (j) add 7

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

Intervention #7-7 SOLVING EQUATION (7-5)

One-Step-Equations Remember, in order to solve basic equations, you must cancel terms by performing the opposite operation on both sides of the equation. Division and multiplication are opposite operations. Addition and subtraction are opposite operations. Here are some examples: (A)

x +3 = 5 x +3−3 = 5−3 x = 2

(B)

x − 4 = 10 x − 4 + 4 = 10 + 4 x = 14

(C) 5x = 25 5x = 25 5 5 x = 5 (D)

x =5 3 x 3· = 5·3 3 x = 15

Directions Show all steps in solving the following equations, then circle your answer and corresponding letter below, revealing the name of one many professional careers that require good math skills. (1) x + 4 = 9

(2) x − 3 = 12

(3) x + 11 = 21

(4) x − 5 = 23

(5) x + 19 = 30

(6) x − 8 = −2

(7) 6x = 24

(8)

x =8 3

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Intervention #7-8

SOLVING EQUATIONS (7-5)

The Two-Step Generally, in solving two-step equations, start by canceling out the number being added to or subtracted from the variable, then cancel the number being multiplied times or divided into the variable. For example: 3x + 2 = 14 3x + 2 − 2 = 14 − 2 3x = 12

1. Cancel out the “2” by subtracting it from both sides

3x 12 = 2 3 x = 4

2. Divide both sides by “3” to cancel out the “3” leaving “x” by itself. Directions

Showing all steps, solve the equations then match your answers with the choices on the right and them to answer the following question: Who invented the Rubik’s Cube? (1) 5x + 8 = 28

(2) 2y − 4 = 8

(4) 3x − 9 = −3

(5)

(3) 4x + 11 = 15

x +1=5 4

A. 5 B. 4 C. −6 I. 6 J. 7 K. 2 R. 16 U. 1 Y. 0

5

152

3

1

2

4

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

153

CHAPTER 12 RAR – POST UNIT TEST (7) PRE-ALGEBRA CONCEPTS

Directions Solve to the following word problems, showing as much work as needed. Calculators are not allowed for this assessment. 7.1 Adding Integers (1) −7 + 12 (2) −15 + (−18) (3) 13 + (−32) (4) −14 + (−14) (5) −42 + 27 7.2 Subtracting Integers (6) −31 − 19 (7) −25 − (−16) (8) 12 − (−16) (9) −15 − (−28) (10) 19 − 40 7.3 Multiplying Integers (11) −6 · 9 (12) 4 · (−12) (13) −7 · (−8) (14) 11 · (−9) (15) −5 · (−5) 7.4 Dividing Integers (16) 54 ÷ −9 (17) −36 ÷ −12 (18) −28 ÷ 7 (19) 40 ÷ −10 (20) −18 ÷ −3 7.5 Determine the value of “n” in each of the following equations. (Show all steps). (21) n + 4 = 22

154

(22) 9n = 45

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS

(23) n − 12 = 31

(24) 7n + 4 = 11

(25) n + 21 = −8

Solution Intervention 7-1 S 0

T −1

A 17

T 8

I −5

S 3

T −9

I 1

C 9

S 10

included data collection, analysis, and interpretation. Solution Intervention 7-2 C A L −21 6 −17

C 5

U −2

L 4

U 15

S 30

Solution Intervention 7-3 “Two Words” (1) −18;

(2) −10; (3) 3;

(8) −25; (9) 12; (15) 58;

(4) −8;

(10) −22;

(5) 38; (6) −19;

(11) −29; (12) −17;

(7) −46;

(13) 6;

(14) −23;

3 (16) 21; (17) −38; (18) −16; (19) 1.4; (20) − . 8

Solution Intervention 7-4 P 12

E 1

R 3

C 8

Y 7

S 11

P 2

E 4

N 6

C 5

E 10

R 9

In 1945, Percy L. Spencer invented the Microwave Oven. Solution Intervention 7-5 (1) −34;

(2) 9;

(8) −25;

(9) −32;

(14) 3;

(3) 7;

(4) 11; (5) 31; (6) −12;

(10) 15;

(11) 16; (12) 10;

(7) −14; (13) −12;

(15) −4.

155

CHAPTER 12

Solution Intervention 7-6 (1) f;

(2) j;

(3) a;

(4) c;

(5) g;

(6) e;

(7) b;

(8) i;

(9) h;

(10) d

Solution Intervention 7-7 ENGINEER (1) 5;

(2) 15;

(3) 10;

(4) 28;

(5) 11;

(6) 6;

(7) 4;

(8) 24

Solution Intervention 7-8 (1) 4;

R 5

(2) 6;

U 3

(3) 1;

B 1

I 2

(4) 2;

(5) 16.

K 4

Erno Rubik invented the Rubik’s Cube, a 3-D mechanical puzzle in 1974. Solution Unit 7 Supplemental Quizzes Quiz 7A: (a) −4; (b) 38; (c) −36; (d) 4; (e) 6. Quiz 7B: (a) 6; (b) −7; (c) 30; (d) −3; (e) 13. Quiz 7C: (a) −46; (b) −8; (c) −72; (d) −4; (e) 50. Quiz 7D: (a) 1.3; (b) 12; (c) 2; (d) −27; (e) 4. Solution – Post Unit Assessment (7) (1) 5;

(2) −33;

(3) −19;

(4) −28;

(5) −15;

(8) 28;

(9) 13;

(10) −21;

(11) −54; (12) −48; (13) 56; (14) −99;

(15) 25; (16) −6; (17) 3; (18) −4; (19) −4; (22) 5; 156

(23) 43;

(24) 1;

(25) −29.

(6) −50:

(20) 6;

(7) −9;

(21) 18;

SPECIFIC CONTENT AREA INTERVENTIONS – PRE-ALGEBRA CONCEPTS GRADING SCALE – POST UNIT ASSESSMENT (7)

157

APPENDIX A POST ALGEBRA READINESS ASSESSMENT – (VERSION 2A) (ARA)

Directions Solve the following problems, showing as much work as needed. Calculators are not allowed for this assessment. I. Whole numbers (1) 10,308 + 709 + 65

[1.1]

(3) 704 × 98

 (4) 3 8022

[1.3]

(2) 2,890 − 897 [1.2] [1.4]

 (5) 37 842

(remainder)

[1.4]

(7) 14.3 − 9.07

[2.2]

II. Decimals (6) 4.13 + 15 + 9.2 (8) 8.07 · 3.9

[2.3]

[2.1]  (9) 4.3 15.48

[2.4]

(10) 983.7 ÷ 1000

[2.4]

III. Fractions (Simplify answers if needed) (11)

3 1 + 4 6

[3.1]

3 1 (12) 4 + 2 5 2

[3.1]

(13)

8 5 − 9 6

[3.2]

5 3 (14) 8 − 6 8 4

[3.2]

(15)

4 5 · 9 6

2 1 (16) 1 · 2 3 4

[3.3]

[3.3]

159

APPENDIX A

(17) 10 ÷

5 9

2 4 (18) 1 ÷ 1 5 5

[3.4]

(20) Write 0.2 as a percent?

[4.2]

[3.4]

IV. Percents (19) Write

3 4

as a percent?

(21) What is 25% of 80?

[4.1] [4.3]

V. Number concepts and basic geometry (22) Find the average of: 75, 201, 39

[5.4]

(23) Write 13.05 in words.

[5.1]

(24) What is the value of n:

n 3 = 8 2

(25) Compute: 12 + 6 · 2

[5.5]

[5.6]

(26) Round 7.718 to the nearest hundredth. (27) What is the area of the rectangular?

[5.3] [5.8]

4 cm 9 cm

(28) Arrange in order from least to greatest: (a) 0.6 (b) 0.06 (c) 0.51 (d) 0.504

[5.2]

VI. Word problems (29) If Elaine’s total purchase at the music store came to $13.73 and she gave the sales clerk a $20 bill, how much change she should she get back? [6.1] (30) As a babysitter, Jorge earned $50.75 for seven hours of work. How much did he earn for each hour that he worked? [6.2]

160

APPENDIX A

VII. Basic integers and pre-algebra (31) What is 12 + (−8)?

[7.1]

(32) What is −12 · (−2)?

[7.2]

(33) What is the value of x: x − 21 = 40

[7.3]

161

APPENDIX B POST ALGEBRA READINESS ASSESSMENT (VERSION 2B-C) (ARA) Directions Solve the following problems, showing your all work on separate sheets of paper. Calculators1 are not allowed for this assessment. When you are finished, attach your work to the back of your answer sheet.

I. Whole numbers (1) 15,069 + 556 + 41

[1.1]

(A) 14,666 (B) 15,766 (C) 15,666 (D) 14,566

(3) 803 × 96

(2) 7,015 − 904

[1.2]

(A) 6,111 (B) 7,111 (C) 6,011 (D) 6,101

[1.3]

(A) 7,788 (B) 77,088 (C) 7,708 (D) 76,088

 (4) 5 835

[1.4]

(A) 1067 (B) 157 (C) 1067 (D) 167

 (5) 69 7709 (remainder) [1.4] (A) 11 R 5 (B) 111 R 5 (C) 110 R 119 (D) 111 R 50

II. Decimals (6) 6.05 + 5.9 + 15 (A) 6.79 (B) 2.695

[2.1]

(7) 18.3 − 7.04

[2.2]

(A) 10.89 (B) 9.99

1 Students with IEPs or 504 plans allowing calculators or other assistive technology may use such devices.

163

APPENDIX B

(C) 26.95 (D) 25.85

(8) 38.7 · 1.9

(C) 11.26 (D) 9.04

[2.3]

(A) 7.353 (B) 73.53 (C) 735.3 (D) 74.53

 (9) 2.5 9.25

(10) 543.7 ÷ 1000

[2.4]

(A) 37 (B) 3.7 (C) 0.37 (D) 0.307

(A) 0.5437 (B) 5.437 (C) 54.37 (D) 543.70

III. Fractions (Simplify answers if needed) (11)

1 4 + 4 5

(A) 1 (B)

[3.1]

1 20

5 12 1 (B) 6 12 6 (C) 6 10 1 (D) 7 12

5 9

1 9 2 (D) 2 20 5 1 − 9 3

2 3 4 (B) 6 2 (C) 9 1 (D) 9 (A)

164

[3.1]

(A) 7

(C) 1

(13)

1 5 (12) 4 + 2 4 6

[3.2]

(14) 13

3 11 −2 10 15

17 30 17 (B) 10 30 8 (C) 11 5 7 (D) 10 30 (A) 11

[3.2]

[2.4]

APPENDIX B

(15)

5 7 · 21 20

1 1 (16) 3 · 2 5 4

[3.3]

1 12 35 (B) 400 1 (C) 7 5 (D) 12

1 5 1 (B) 6 20 1 (C) 5 15 36 (D) 20

(A)

(17) 8 ÷

[3.3]

(A) 7

2 9

[3.4]

(18)

25 1 ÷8 36 3

[3.4]

125 108 (B) 12

(A) 18

(A)

(B) 36 16 9 7 (D) 1 9

1 12 5 (D) 8 36

(C)

(C)

IV. Percents (19) Write

1 5

as a percent.

[4.1]

(A) 55% (B) 25% (C) 50% (D) 20% (20) Write 0.9 as a percent.

[4.2]

(A) 0.09% (B) 900% (C) 90% (D) 0.9%

165

APPENDIX B

(21) What is 40% of 50?

[4.3]

(A) 2 (B) 20 (C) 200 (D) 0.2

V. Number concepts and basic geometry (22) Find the average of: 99, 24, 801

[5.4]

(A) 308 (B) 924 (C) 307 (D) 300 (23) Write 9.04 in words.

[5.1]

(A) nine and four hundredths (B) nine and four tenths (C) nine and four thousandths (D) nine point four (24) What is the value of x:

6 9 = ? 2 x

[5.6]

(A) 18 (B) 3 (C) 6 (D) 1 (25) Compute: 24 − 5 · (2 + 1)

[5.5]

(A) 15 (B) 18 (C) 9 (D) 57 (26) Round 9.054 to the nearest hundredth. (A) 9.1 (B) 9.051 (C) 9.06 (D) 9.05

166

[5.3]

APPENDIX B

(27) What is the perimeter of the rectangular?

[5.7]

8 cm (A) 64 cm (B) 32 cm (C) 16 cm (D) 28 cm (28) Arrange in order from least to greatest: (a) 0.09

(b) 0.1

(c) 0.02

(d) 0.201

[5.2]

(A) c, a, b, d (B) b, c, d, a (C) c, d, b, a (D) d, a, c, b

VI. Word problems (29) The Lopez’s family dinner at their favorite restaurant came to a total of $70. If they want to leave the waiter a 15% tip, how much tip should they leave him? [6.1] (A) $11 (B) $10 (C) $7 (D) $10.50 (30) Marlow wants to bike 95 miles over 3 days. He bikes 31 miles on the first day and 38 miles on the second. How many miles must he bike on the third day to reach his goal? [6.3] (A) 69 miles (B) 16 miles (C) 26 miles (D) 36 miles

167

APPENDIX B

VII. Basic integers and pre-algebra (31) What is −23 + 32?

[7.1]

(A) 9 (B) −9 (C) −53 (D) 53 (32) What is −15÷ 3?

[7.2]

(A) −12 (B) −5 (C) 5 (D) 45 (33) What is the value of x: 2x + 4 = 14 (A) −5 (B) 5 (C) −10 (D) 10

168

[7.3]

ANSWER KEY POST ALGEBRA READINESS ASSESSMENT (ARA)

169

ANSWER KEY GRADING SCALE – ALGEBRA READINESS ASSESSMENT (ARA)

170

REFERENCES

Adelman, C. (1999). Answers in the tool box: Academic intensity, attendance patterns, and bachelor’s degree attainment. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Achieve, Inc. (2006). Closing the Expectations Gap, Washington, D.C., www.achieve.orgww.ac. Business Higher Education Forum (2005). A ommitment to America’s future: Responding to the crisis in mathematics and science education – The main report. Washington, D.C., http://www.bhef.com/ www/publications/documents/commitment\_future\_05.pdf. Carnevale, A. P. & Desrochers, D. M. (2003). Preparing students for the knowledge economy: What school counselors need to know. Professional School Counseling, 6, 228–236. Cotton, K. & Wikelund, K. R. (1989). Parent Involvement in Education. NW Regional Educational Laboratory, http://www.nwrel.org/scpd/sirs/3/cu6.html. Evan, A., Gray, T., & Olchefske, J. (2006). The gateway to student success in mathematics and science. Washington, D.C.: American Institutes for Research. Horn, L. & Nuñez, A.-M. (2000). Mapping the road to college: First-generation students’ Math track, planning strategies, and context of support. Washington, D.C.: U.S. Department of Education, National Center for Education Statistics. National Mathematics Advisory Panel (Spring 2008). Final Report, Washington, D.C. National Commission on Excellence in Education (1983). A nation at risk: The imperative for education reform, Washington, D.C. Williams, T. (2008a). 100 Algebra Workouts and Practical Teaching Tips. Carthage: Teaching and Learning Company. Williams, T. (2008b). 100 Math Workouts and Practical Teaching Tips. Carthage: Teaching and Learning Company.

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